Lec 06 - Relations

Summary

This video script explores the concept of sets and relations in mathematics. It begins by explaining set operations like union, intersection, and difference, and introduces set comprehension for creating subsets. The script then delves into the Cartesian product, a method of combining sets to form pairs, and uses it to define binary relations. Examples of relations, both numerical and conceptual, are given, such as teacher-course allocations and parent-child relationships. The video also covers properties of relations, including reflexivity, symmetry, transitivity, and anti-symmetry, culminating in the explanation of equivalence relations and equivalence classes. The script effectively uses geometric visualizations and real-world analogies to clarify abstract mathematical ideas.

Takeaways

• 😀 Sets are collections of items from which new sets can be constructed through operations like unions, intersections, and differences.
• 📚 Set comprehension is a notation that allows for creating subsets based on a base set and a specified condition.
• 🤝 The Cartesian product is a method of combining two sets to form a new set consisting of ordered pairs from each set.
• 📏 In the Cartesian product, the order of elements is crucial, meaning that (a, b) is not the same as (b, a).
• 📈 Relations are subsets of a Cartesian product, formed by applying a condition to select specific pairs, which can be visualized as points on a graph.
• 🔍 Relations can be represented in various ways, including as a graph with nodes and arrows, or by using set-builder notation.
• 🔢 Relations can define geometric shapes, such as a circle, by specifying conditions on the elements' coordinates.
• 🏷️ Special binary relations include the identity relation, which maps every element to itself, and equivalence relations, which are reflexive, symmetric, and transitive.
• 🔄 Equivalence relations partition a set into disjoint equivalence classes, where elements within a class are considered equivalent, and elements across different classes are not.
• 📊 Relations can be defined on an arbitrary number of sets, not just binary relations, and can involve tuples of varying lengths.
• 📋 Properties of relations such as reflexivity, symmetry, transitivity, and anti-symmetry are important for understanding the nature and behavior of the relationships between elements.

Q & A

What is a set in the context of this script?

A set is a collection of items from which new sets can be constructed through operations like unions, intersections, and differences.

What is set comprehension notation used for?

Set comprehension notation is used to carve out subsets of a set by applying some condition to the elements of the base set and collecting the elements that meet the condition.

Can you explain the concept of the Cartesian product in the context of sets?

The Cartesian product is a way to combine two sets to form a new set consisting of all possible ordered pairs, where the first element of each pair comes from the first set and the second element comes from the second set.

How is the order of elements important in the context of pairs in a Cartesian product?

In the context of pairs in a Cartesian product, the order of elements is crucial because it determines the distinctness of pairs. For example, the pair (a, b) is not the same as the pair (b, a) unless a equals b.

What is a relation in terms of sets and the Cartesian product?

A relation is a subset of a Cartesian product that is formed by applying a condition to filter out pairs of interest from the set of all possible pairs.

How can a relation be denoted or represented?

A relation can be denoted by naming it as a set and stating that a pair (a, b) belongs to the relation R, or by using the relation as an operator, writing a R b to indicate that a is related to b by R.

Can you provide an example of a relation outside the context of numbers?

An example of a relation outside the context of numbers is an allocation relation in a school, where the set of teachers T is crossed with the set of courses C to form pairs indicating which teacher is teaching which course.

What is an equivalence relation and how does it differ from other relations?

An equivalence relation is a special type of binary relation that is reflexive, symmetric, and transitive. It partitions a set into disjoint equivalence classes, where elements within a class are equivalent to each other and not equivalent to elements in other classes.

How does the script explain the concept of equivalence classes?

The script explains equivalence classes as groups of elements that are equivalent to each other based on an equivalence relation. It uses the example of integers modulo 5, which partitions the integers into five disjoint classes based on their remainder when divided by 5.

What are the properties of binary relations mentioned in the script?

The script mentions reflexivity, symmetry, transitivity, and anti-symmetry as properties of binary relations. These properties help define the nature of the relation between elements in a set.

Can you give an example of how relations can define geometric shapes?

The script provides an example where all points (a, b) in the Cartesian product R x R that are at a distance of 5 from the origin (0, 0) define a circle with a radius of 5 centered at the origin. This demonstrates how relations can be used to describe geometric shapes.

Outlines

00:00

🔍 Introduction to Relations and Set Operations

The paragraph introduces the concept of relations, building upon the idea of sets as collections of items. It discusses how new sets can be formed through operations like unions, intersections, and differences, and how set comprehension is used to create subsets based on certain conditions. The Cartesian product is introduced as a method to combine two sets by forming pairs of elements, emphasizing the importance of order in these pairs. The concept is illustrated with examples, including the set of integers divisible by 2, fractions with coprime numerators and denominators, and real numbers within specific intervals. The paragraph also explains how to visualize these relations as points on a graph, particularly when dealing with numbers.

05:04

📚 Relations in Various Contexts and Their Representation

This paragraph delves into different contexts where relations can be applied, such as in a school setting where teachers and courses are represented as sets, and the allocation relation shows which teachers are teaching which courses. It also discusses the representation of relations as graphs with nodes and arrows, and how this can be used to visualize relations on finite sets. The paragraph further explores relations in the context of familial relationships, specifically between mothers and children, and how geometric shapes can be defined by relations, such as points at a certain distance from the origin forming a circle. It also touches on how relations can be viewed in different ways, such as rational numbers in reduced form being a relation on integers.

10:07

📐 Advanced Relations and Geometric Interpretations

The paragraph discusses the extension of the Cartesian product notation to multiple sets and provides examples, including Pythagorean triples that satisfy the theorem a^2 + b^2 = c^2, and squares on a plane defined by their corners. It explains how these relations can involve multiple sets of points, each being a pair of real numbers. The concept of n-tuples is introduced, showing how relations can be defined on an arbitrary number of copies of a set, moving from pairs to triples to quadruples, and so on. Special binary relations such as the identity relation, which maps every element to itself, are also highlighted, with examples of how they can be simplified in notation.

15:12

🔄 Properties of Relations: Reflexivity, Symmetry, and Transitivity

This paragraph explores various properties that relations may possess, such as reflexivity, where an element is related to itself, symmetry, where the relation of a to b implies the relation of b to a, and transitivity, where a relation between a and b and between b and c implies a relation between a and c. It provides examples of these properties, such as the divisibility relation being reflexive, the greatest common divisor relation being symmetric, and the less than relation being transitive. The paragraph also introduces anti-symmetry, which states that if a relation exists from a to b, then it cannot exist from b to a in the opposite way, using the mother-child relationship as an example.

20:17

🔗 Equivalence Relations and Their Impact on Set Partitioning

The final paragraph introduces equivalence relations, which are relations that are reflexive, symmetric, and transitive. It explains how equivalence relations can partition a set into disjoint classes, where all elements within a class are equivalent to each other, and all elements across different classes are not equivalent. The concept of modulo arithmetic is used as an example to illustrate how equivalence relations can group numbers with the same remainder when divided by a certain number. The paragraph also relates equivalence relations to the concept of a clock, which partitions time into AM and PM, showing how equivalence classes can represent a form of equality with similar properties to actual equality.

Keywords

💡Set

A set is a fundamental concept in mathematics representing a collection of distinct items, which can be numbers, objects, or even other sets. In the video, sets are used as the basis for constructing more complex mathematical structures. For example, new sets can be formed by taking unions, intersections, and differences of existing sets, which are operations that combine sets in various ways to create new ones.

💡Relation

A relation in mathematics is a way of describing a connection between two sets. The video script introduces relations as subsets of a Cartesian product, which are formed by applying certain conditions to pairs of elements from two sets. Relations are essential in understanding how elements from different sets can be associated with one another, and they can be visualized using graphs or other means.

💡Cartesian Product

The Cartesian product is a operation that combines two sets to create a new set, where each element of the new set is an ordered pair consisting of one element from each of the original sets. In the video, the Cartesian product is used to form pairs that can then be filtered by certain conditions to define relations. For instance, the script mentions 'A cross B' to denote the set of all ordered pairs (a, b) where a is from set A and b is from set B.

💡Set Comprehension

Set comprehension is a concise way to define a set by specifying a rule or condition that its elements must satisfy. In the video, set comprehension is used to describe subsets of a base set that meet certain criteria, such as all integers divisible by 2 or fractions where the numerator and denominator have no common divisors. This concept is crucial for creating specific sets that are relevant to the discussion of relations.

💡Binary Relation

A binary relation is a specific type of relation that involves two sets. It is a subset of the Cartesian product of these two sets, where only certain pairs that meet a given condition are included. The video script discusses binary relations extensively, using them to illustrate various mathematical concepts, such as the teacher-student relation or the mother-child relation.

💡Equivalence Relation

An equivalence relation is a special type of binary relation that is reflexive, symmetric, and transitive. It is used to partition a set into disjoint equivalence classes, where elements within each class are considered equivalent based on the relation's criteria. In the video, equivalence relations are exemplified by the relation of congruence modulo 5, which groups integers based on their remainder when divided by 5.

💡Reflexivity

Reflexivity is a property of a relation where every element is related to itself. This means that for a relation to be reflexive, each element a in the set must be paired with itself (a, a) in the relation. The video script uses the example of the identity relation on natural numbers to illustrate reflexivity, where every number is paired with itself on a grid.

💡Symmetry

Symmetry in the context of relations means that if one element is related to another, then the second element is also related to the first. In other words, if (a, b) is in the relation, then (b, a) must also be in the relation. The script gives the example of the greatest common divisor relation, where if 'a' and 'b' share no common divisors, the same is true if 'b' and 'a' are considered.

💡Transitivity

Transitivity is a property of relations that states if 'a' is related to 'b' and 'b' is related to 'c', then 'a' must also be related to 'c'. This property is important for understanding how relations can propagate through a set. The video script explains transitivity with the example of divisibility, where if a number 'a' divides 'b', and 'b' divides 'c', then 'a' divides 'c'.

💡Anti-symmetry

Anti-symmetry is a property of a relation where if (a, b) is in the relation, then (b, a) cannot be in the relation. This property is used to ensure that the relation does not allow for two elements to be related in both directions. The video provides the example of the 'less than' relation, where if 'a' is less than 'b', then 'b' cannot be less than 'a'.

Highlights

Introduction to the concept of Relations as a way to combine sets into new sets.

Explanation of set operations such as union, intersection, and difference.

Set comprehension notation for creating subsets based on conditions.

Introduction of Cartesian product for forming pairs from two sets.

Visual representation of Cartesian products as grids or graphs.

Definition and example of binary relations using set comprehension.

Different notations for expressing relations, such as R(a, b) and aRb.

Real-world application of relations in teacher-course allocation.

Graphical representation of relations using nodes and arrows.

Relation between mothers and children as an example of a binary relation.

Use of Pythagorean theorem to define relations in geometry.

Concept of n-tuples and their relation to Cartesian products.

Special binary relations like identity, and their properties.

Properties of relations: reflexivity, symmetry, and transitivity.

Explanation of anti-symmetry and its difference from symmetry.

Introduction and characteristics of equivalence relations.

Equivalence relations partitioning sets into disjoint equivalence classes.

Practical example of equivalence relations using a clock for modulo arithmetic.

Summary of the key concepts and properties of relations discussed in the transcript.

Transcripts

So, we have seen Sets, now let us move on to Relations. As we saw a set is a collection

of items and we can construct new sets from old sets. So, we can take unions combine two

sets into one. We can take intersections, take the common elements. We can take the

difference that is take the elements of X which are not in Y and if we define the universe

with with respect to which we are working, we can define the complement those elements

that are not in X. Now, in general we are interested in carving

out subsets of a set and so, we use the set comprehension notation. So, what this does

is it takes a base set and takes elements of that set, then it applies some condition

those elements we are interested in and then it collects them all together. So, we can

take all the integers which are divisible by 2 or not divisible by 2 in this case, so

we get the odd one; so, those where the remainder is 1.

Or we can take all fractions in which the numerator and the denominator have no common

divisor or we can take for instance the real numbers which lie in an interval in the with

the left closed interval 3 and the right open interval 17.

So, now, we will see a new way to combine sets to form new sets and this is called the

Cartesian product. And, in the Cartesian product basically what we do is we take two sets and

we take one element from each and form a pair. So, A cross B as it is called is the set of

all pairs which we write with this normal bracket notation a comma b such that the first

element A comes from the big the set capital A and the second element comes from the set

capital B . So, for instance, if A is the set 0, 1 and

B is a set 2, 3 then all possible pairs we can form in the Cartesian product a 0 combined

with 2. So, 0 comma 2, 0 comma 3 and then 1 combined with 2, 1 comma 2 and 1 comma 3.

So, we have four possible pairs . Now, in sets we said that the order of the

element is not important, but of course, when we are doing this kind of a pairing, then

we know that the left set comes from the left part of the product and the right element

comes from the right part of the product . So, for example, 0 comma 1 is not equal to 1 comma

0. So, here we have to respect the order when we talk about a pair.

Now, if we have sets of numbers right then we normally visualize the product as a space

which we draw familiarly as a graph. So, for instance if we take N cross N then we draw

N cross N as this grid, where on the x-axis you have one copy of N, on the y-axis you

have another copy of N . And, for example, if you want to look at the pair 2 comma 3,

then such that the x-coordinate is 2 and the y-coordinate is 3 and you get this point and

similarly, if you look at the point 5 comma 6 you get this point right.

So, you can take the first coordinate plot it on the x-axis; take the second coordinate

plot it on the y-axis and where those two points meet in the grid is the point that

we are interested in. So, this is one way of visualizing a binary relation on numbers.

And, we can do the same thing if you are using say the reals, in which case the grid points

that we are going to plot will have real coordinates and not just natural number coordinates.

So, now we have this Cartesian product which consists of all possible pairs of the two

sets and as we did with set comprehension we might want to pick out some of these sets

some of these pairs and this is what we call a relation.

So, we combine this Cartesian product operation with set comprehension. So, for instance,

we can take all pairs of numbers which are natural numbers n comma n, but we want to

insist that the second number is 1 plus the first number.

So, we get for instance 0, 1 because the second number one is 0 plus 1; 2, 3 because 3 is

2 plus 1, 17, 18 and so on . And, if we plot these points alone on the right then we get

these so, we get a subset of the overall points and this these points satisfied this set comprehension

condition . Another example would be pairs again of natural

numbers d comma n, where d is a factor of n. Remember, d is a factor of n means that

if I divide n by d, I get remainder 0. So, for instance 2 is a factor of 82, 14 is a

factor of forty 56. So, these will be points in our relation. So, this is what is called

a binary relation. So, formally it is a subset of the product. So, we take the Cartesian

product all possible pairs and then we apply some kind of a condition which filters out

the pairs of interest to us and it gives us therefore, a subset of pairs and this is what

we call a relation. Now, to denote the pairs that belonged to

the relation either we can give the name of the relation as a set and say that a comma

b belongs to R or sometimes to say that a is related to b we use R as a kind of operator.

We say a is related by R to b and so, we write a R b. So, these are two notations which you

might see in different books and they mean exactly the same thing.

So, let us let us look at some other examples of relations outside the numbers. So, supposing

you have a school in which there are some teachers and some courses to be taught. So,

T is the set of teachers; C is the set of courses that are being offered in this term,

then you need to describe which teachers are teaching which courses. So, we would have

an allocation relation A which is a subset of all possible pairs T cross C.

So, every teacher and principle could be teaching every course, but of course, this is not normally

the case. We do not have all teachers teaching all courses, we have some teachers teaching

some courses. So, we would specifically say take every pair of possible teacher course

pairs, then we take out those were precisely the teacher T is actually teaching the course

C and we collect those together to form this allocation relation.

So, here is a different graphical way of describing a relation not in terms of the grid and the

graph that we have learned when we do graphs in school. So, this is also called a graph,

but this is a graph in which we have some nodes representing the elements on the set.

So, on the left hand side we have five teachers, on the right hand side we have 5 four courses

and the arrows from the left hand side to the right hand side connect the pairs which

are in the relation. So, we see that Kumar teaches maths; Deb teaches history and so

on. So, this is a useful useful way of visualizing

relations on finite sets and we will see this often as we go along. Another example of a

similar type of a relation is that between a parent and a child specifically let us look

at mothers and children. So, if we have a set of people in a country, then we can take

the set of all pairs of people and then isolate from that pairs in which the first element

of the pair is the mother of the second element. So, we want m comma c which belongs to P cross

P such that m is the mother of c. So, let us go back to a numbers. So, supposing

we want to plot all points which are in R cross R which are at a distance 5 from 0,

comma 0 which is normally called the origin. So, one thing you need to know for this we

probably you should have learned this at some point is that if I take a point a comma b

and calculate its difference from 0 comma 0. So, this is calculated using the Pythagoras

theorem and it comes out to be square root of a squared plus b squared.

So, in other words, the relation we are looking for in this case is all a comma b whose distance

from 0 comma 0 is 5. So, all a comma b and R cross R such that the square root of a squared

plus b squared is equal to 5. So, here are some of the points 0 comma 5 for instance

you can see 0 comma 5 is there because the sum is 0 plus 25 and the square root of that

is 5. 3 comma 4 is there because 3 squared is 9, 4 squared is 16, 9 plus 16 is 25, square

root to 25 is again 5. So, interestingly these points if we plot

every such point in R cross R which satisfies this actually defines a circle of radius 5

with center at 0 comma 0. So, relations can define interesting geometric shapes and very

often we do deal with geometric shapes in this relational form because it is easier

to manipulate than looking at pictures. Now, depending on how we are going to view a relation,

we can look at it in different ways. So, remember that we looked at rationals in

reduced form. So, we said that a rational in reduced form has p comma q, p by q such

that p and q are integers and the gcd is 1 right; that means, that they do not have a

greatest common divisor other than 1. But, we can also think of this as a relation on

integers itself. We want all pairs of integers. So, every rational is really a pair of integers,

the numerator and the denominator and we want every pair of integers where the gcd is 1,

that is, there is no common divisor. So, we do not have to restrict our self to

binary relations. The Cartesian product notation extends to multiple sets. Let us look at three

sets for instance. Remember, Pythagoras theorem which says that the square on the hypotenuse

is the sum of the squares on the opposite sides. So, what values of a, b and c could

be the sides of a right triangle are determined by Pythagoras's theorem.

So, we would say that a, b and c is a valid triple in the Pythagoras sense if a, b, c

belongs to N cross N cross N. So, here we now have three copies of N and a, b, c must

all be nonzero. They must all be positive length, we do not want to have triangles in

which one line one side is collapsed to a point and we want the constraint that a squared

plus b squared is equal to c squared . Here is another example. Suppose, we look

at squares on the plane squares with real corners right. So, a corner is a point x comma

y which is in R cross R. So, we define the x coordinate and the y coordinate that defines

the corner of a square and we want four corners which together form a square if we connect

them by lines. So, for instance, if you look on the right the four blue dots correspond

to a square which is cornered at 0, 0; 0, 2; 2, 0 and 2, 2.

The red square is also red points also define a square because this is a rotated square,

but then if you rotate it vertically you will turn out that this diamond is actually a square

. So, there are many such four sets of points which form the corners of squares and we might

be interested in all such four sets of points. So, now, we have a relation which involves

four sets of points, but each point itself is a pair of real numbers; it is an x and

a y. So, square if we think of it as a relation

is actually a relation on R squared that is the first corner times R squared the second

corner times R squared the third corner and the fourth corner R squared again. So, this

is actually either a relation on eight copies of R or if you want to group it four copies

of pairs of R . So, this just says that we can take relations

on arbitrary an arbitrary number of copies of a set and we get larger and larger from

pairs we move to triples we move to quadruples and in general if we have n copies we call

this an n tuple . So, there are some special binary relations

which pop up all over the place. So, it is useful to know their names. The first one

is called the identity relation and as you would expect, the identity relation maps every

element to itself. So, if I take A cross A, so, first of all the identity relation is

defined on two copies of the same set because identity means equality. So, I take A cross

A. So, this has all kinds of pairs a comma b, where both a and b belong to A and now,

I want the condition that a is equal to b. So, in other words, I want things of the form

a comma a. So, if I plot this for instance on the natural numbers and N cross N, then

I get 0, 0; 1, 1 and so on, and these are the points which are drawn on the right in

this grid . Now, point of notation we sometimes it is

tedious to write this notation as it is says us a comma b time in n. So, we do not want

to you know have to write out this long thing. So, sometimes we simplify this by saying I

want all pairs such that a comma a belongs to A cross A. So, what we are really saying

is that the second day and the first day must be the same. So, we are collapsing the equality

and this. Now, this is not technically correct, but this is often used in order to simplify

the notation . And, sometimes we might drop the product altogether.

We might just say we want all pairs a comma a where a comes from the set A. So, in other

words we are pulling out one copy of the element from the set and then we are constructing

a pair by taking two copies of it. So, all of these are equivalent ways of writing this

although only the first one technically follows the notation that we are using to introduce

relations . Now, there are some properties that relations

may have. The first one is called reflexivity. So, reflexivity refers to the fact that an

element is related to itself. So, a reflexive relation is one in which for every element

a; a comma a belongs to R. So, in other words based on what we just wrote above, it means

that the identity relation is included in R. So, it does not mean that that is the only

thing. The identity relation has only the reflexive elements. A relation that is reflexive

will have the identity pairs and it will have other pairs, but it must have all the identity

pairs to be called reflexive . A symmetric relation for instance is one where

if a, b is there, then b a must be there. So, for instance looking at reflexive relations

one example is the division relation. So, if we provided we make sure that the numbers

are not 0, then we know it is reflexive because every number divides itself. So, if we take

the reflect division relation as the relation that we introduced in the first part of this

lecture that would be reflexive because a divides a for every a which is not 0.

Similarly, symmetric relations if we look at pairs where the greatest common divisor

is 1, in other words they have no common divisors. This is what happens for example, in reduced

fractions, then it does not matter whether we write it as a comma b or b comma a. So,

if a comma b has greatest common divisor 1, so does b comma a. So, a comma b and b comma

a must both either be there in the relation or neither will be there.

Similarly, if we look at this which is asking about the absolute value so, it is saying

give me all numbers a and b such that a minus b is either plus 2 or minus 2. So, the absolute

value takes the difference and removes the negative sign. Now, we see that for instance

if 5 comma 7 is there, then 7 comma 5 must be there because they both have the same difference

depending on how we write it. Normally, in subtraction we have a sign difference, but

because we are taking the absolute value there is no difference actually between these two

. So, this absolute value relation also if we fix is a symmetric relation.

A third property that relations may have and which are useful is called transitivity. So,

transitivity says that if we have two pairs which are related such that they share an

elements. So, a is related to b and b is related to c, then a must be related to c. So, again

our divisibility is a relation. So, supposing we say that 2 divides 6 and we say that 6

divides 36, then from this we can conclude that 2 divides 36 as well, right.

Similarly, if we take less than if we say that 3 is less than 10 and 10 is less than

28, then we know from this that 3 must be less than 28. So, this is transitivity.

So, if we want to draw it pictorially if we have three elements a and a, b and c and this

arrow remember we had this graph notation which says a is related to b and b is related

to a, then this dashed line represents the requirement for transitivity a must be related

to c . Now, we saw symmetry. So, symmetry says that

if a, b is in R, then b comma a must also be in R. Anti-symmetry says something different

it says if a, b is in R, then b comma a should not be in R. So, less than for example, which

was transitive above is also anti-symmetric. If you take strictly less than, if a is strictly

less than b then it cannot be that be strictly less than a. So, this is an anti symmetric

relation, but anti symmetry does not require that one of the two must be there. It only

says that if one pair is there the opposite pair should not be there ok.

Similarly, if we look at our mother and children example; obviously, if p is the mother of

c then c cannot be the mother of p ok. Now, there may be p and c such that neither p is

the mother of c nor is c is the mother of p. So, that is allowed. We do not insist that

every pair p comma c must be related one way or another, but if it is related one way it

should not be related the other way is what anti-symmetry says.

So, if we combine some of these conditions we get an interesting class relations called

equivalence relations. So, equivalence relation is something that is reflective, reflexive,

symmetric and transitive. So, as an example supposing we connect together all numbers

which have the same remainder modulo 5. So, for instance 7 has a remainder 2 with respect

to 5 and so does 22. So, 7 and 5 would be related in this way if we define the relationship

as having the same remainder modulo 5. Now, notice that if two numbers have the same

remainder modulo 5; that means, that going from one number to the other you are going

in multiples of 5. So, for instance 22 minus 7 is 15 right. So, this is this modulo arithmetic.

So, if you add the number that you are dividing by, then you get the same remainder and so,

in set notation we can say that the integers modulo 5 are all pairs a, b such that b minus

a mod 5 is 0. In other words, we are not asking what is the actual remainder of b and a, we

are just saying that b and a are separated by a multiple of 5 therefore, they must have

the same remainder modulo 5. Now, this divides the integers into five groups

if I based on the remainder. So, there are the group of numbers which are divisible by

5, they have remainder 0. Those like 6, 11 and all which have remainder 1, 7, 12 and

all which one remainder 2 and so on. So, we have five possible remainders 0, 1, 2, 3,

4 and therefore, this divides the set of integers into five disjoint classes.

As an example of modulo arithmetic that we are all familiar with, consider what happens

when we look at a normal clock. Now, a normal clock measures time from 0 to 12 and then

cycles around again. So, though there are 24 hours in a day, the clock is actually partitioning

these 24 into two sets where we have 0 and 12 as same, 1 and 13 as same and so on right.

So, 2 am and 2 pm there is no distinction on the clock .

So, the clock is actually showing us this equivalence class of hours regarding am and

pm as being equal and we have to know from context whether the clock is showing am or

pm. So, the main thing to note about an equivalence relation is that it partitions a set. It partitions

a set into disjoint groups, all of the elements within a group are equivalent and all of the

elements outside across groups are not equivalent to each other.

So, the groups of equivalent elements that we formed through an equivalence relation

are called equivalence classes . So, this might look a little abstract now, but equivalence

classes really represent a kind of equality and sometimes we are happy to work with this

equality in terms of equivalence relations rather than actual equality and it has very

much the same properties as equality does .

So, to summarize as we have seen a Cartesian product can generate n-tuples of elements

from n sets. So, if we have x 1, x 2, up to x n n n sets these can be different or the

same then we can take one element from each set and form an n-tuple small x 1, small x

2 up to small x n . And, when we now pick out some particular subset of these n-tuples

we get a relation. So, for instance, if we take pairs from N cross R and we want the

second element of the pair the real number to be the square root of the first element,

then we get N cross R such that r is square root of m .

So, here on the right we have seen we show one picture of this. So, there are some elements

like 2 comma square root of 2, 4 comma 2, 7 comma square root of 7 and so on. Now, just

notice that in this picture the y-axis is elongated compared to the x-axis. So, this

is not in some sense to scale in both dimensions because the square root function behaves like

this. So, we have seen that there are some properties

that we would like to record of binary relations - reflexivity, symmetry, transitivity and

sometimes anti-symmetry. And, using reflexivity, symmetry and transitivity together we get

what is called an equivalence relation an equivalence relations partition sets into

equivalence classes which behave like equality. Thank you .