Types of sets
Summary
TLDRThis educational script introduces various types of sets in set theory, including the empty set, singleton set, finite and infinite sets, equal sets, equivalent sets, universal set, subset, proper subset, superset, and proper superset. It explains the concept of cardinality and provides examples for each type. The script also covers the calculation of subsets and proper subsets using formulas, and defines the power set as the collection of all subsets of a given set. It uses practical examples to illustrate these concepts, making it accessible for learners.
Takeaways
- 😀 An **empty set** is a set with no elements, denoted by the symbol Ø or the term 'null set', and has a cardinality of zero.
- 🎯 A **singleton set** is a set containing only one element, with a cardinality of one.
- 🌐 A **finite set** has a limited number of elements, such as the set of counting numbers less than 6.
- 🌌 An **infinite set** contains an unlimited number of elements, like the set of all counting numbers.
- 🔄 **Equal sets** are sets that have the exact same elements, even if the elements are arranged differently.
- 🔢 **Equivalent sets** have the same number of elements but with different elements, like the sets of letters from different words.
- 🌍 The **universal set** is a set that contains all elements under consideration, with all other sets being its subsets.
- 📚 A **subset** is a set where every element of one set (A) is also an element of another set (B).
- 📘 A **proper subset** is a subset where all elements of set A are in set B, but set A is not equal to set B.
- 📙 A **superset** is a set that contains all elements of another set, and possibly additional elements.
- 📗 A **proper superset** is a superset that is not equal to the original set, meaning it has at least one element not in the original set.
- 🔑 The **power set** of a set is the set of all possible subsets, including the empty set and the set itself, with the number of elements calculated as two to the power of the number of elements in the original set.
Q & A
What is an empty set?
-An empty set, denoted by the symbol ∅ or set e, is a set with no elements. Its cardinality is equal to zero.
How is a singleton set defined?
-A singleton set is a set that contains exactly one element. The cardinality of a singleton set is one.
What distinguishes a finite set from an infinite set?
-A finite set has a limited number of elements, whereas an infinite set has an unlimited number of elements.
What does it mean for two sets to be equal?
-Two sets are equal if they contain exactly the same elements, regardless of the order or form in which they are presented.
What is the difference between equivalent sets and equal sets?
-Equivalent sets have the same number of elements but not necessarily the same elements, while equal sets have the same elements.
What is the role of a universal set in set theory?
-A universal set is a set that contains all elements under consideration in a particular problem, and all other sets in that context are subsets of the universal set.
How can you determine if set A is a subset of set B?
-Set A is a subset of set B if every element of set A is also an element of set B.
What is the formula to calculate the number of subsets for a set with 'n' elements?
-The number of subsets for a set with 'n' elements is calculated using the formula 2^n.
How do you define a proper subset?
-A set A is a proper subset of set B if all elements of A are in B, but A is not equal to B, meaning B has at least one element not in A.
What is a superset and how does it relate to a subset?
-A superset is a set that contains all elements of another set. It is the reverse concept of a subset, where a subset is contained within the superset.
How is the power set of a set defined, and how many elements does it have?
-The power set of a set is the set of all possible subsets of that set, including the set itself and the empty set. The number of elements in the power set is 2^n, where n is the number of elements in the original set.
Outlines
🔢 Set Theory Basics
This paragraph introduces fundamental concepts in set theory. It begins with the definition of an empty set, denoted by the symbol for set E or the null set, and explains that its cardinality is zero. It then moves on to the singleton set, which contains only one element, and illustrates this with set S. Finite sets are exemplified by set F, which includes the first five counting numbers, while infinite sets are represented by set N, encompassing all counting numbers. The concept of equal sets is introduced through set E, which contains letters from the word 'earth,' and set H, which contains letters from 'heart,' showing that despite different elements, they are equal because they have the same number of elements. The paragraph also explains equivalent sets, which have the same number of elements but different elements, using set F and set H as examples. It introduces the universal set, which contains all elements and of which all other sets are subsets. The concept of subsets and proper subsets is also discussed, with examples provided to illustrate these relationships. The paragraph concludes with a discussion on the number of subsets a set can have, using the formula 2^n, where n is the number of elements in the set.
📚 Advanced Set Relationships
The second paragraph delves into more complex relationships between sets. It starts by defining a proper subset, where one set is a subset of another but is not equal to it, using sets A, B, C, and D as examples. The paragraph then explains how to calculate the number of proper subsets using the formula 2^n - 1, where n is the number of elements in the set. It also introduces the concept of a superset, which is a set that contains all elements of another set, and a proper superset, which is a superset that is not equal to the original set. The paragraph concludes with an explanation of the power set, which is the set of all subsets of a given set. It provides the formula for calculating the number of elements in a power set, which is 2^n, where n is the number of elements in the original set. An example is given to illustrate the calculation of the power set for a set with three elements.
Mindmap
Keywords
💡Empty Set
💡Singleton Set
💡Finite Set
💡Infinite Set
💡Equal Sets
💡Equivalent Sets
💡Universal Set
💡Subset
💡Proper Subset
💡Superset
💡Proper Superset
💡Power Set
Highlights
Definition of an empty set, a set with no elements, denoted by the symbol for set E.
The cardinality of an empty set is zero, representing the number of elements.
A singleton set, defined as a set with exactly one element, exemplified by set S.
Cardinality of a singleton set is one, indicating a single element.
Finite set described as a set with a limited number of elements, illustrated by set F containing counting numbers less than 6.
Infinite set characterized by having unlimited elements, such as the set of counting numbers.
Equal sets are those that contain the same elements, demonstrated by sets E and H with identical letters.
Equivalent sets have the same number of elements but different elements, like sets F and H with a cardinality of 5 each.
Universal set defined as the set containing all elements, of which all other sets are subsets.
Subset relationship where set A is a subset of set B if every element of A is also in B.
Proper subset defined as a subset where set A is a proper subset of B if all elements of A are in B but not all elements of B are in A.
Superset relationship where set A is a superset of set B if all elements of B are in A.
Proper superset where set A is a proper superset of set B if A contains all elements of B and additional elements.
Power set defined as the set of all subsets of a given set, exemplified by the power set of set S.
Formula to calculate the number of subsets: 2 to the power of n, where n is the number of elements in the set.
Formula to determine the number of proper subsets: 2 to the power of n minus 1, excluding the set itself.
Explanation of how to list all proper subsets of a set, including those with two, one, or zero elements, and the empty set.
Calculation of the number of elements in the power set using the formula 2 to the power of the number of elements in the original set.
Transcripts
types of sets empty set singleton set
finite set infinite set equal sets
equivalent sets universal set subset
proper subset superset proper superset
powerset empty set a set with no element
this is the symbol of an empty set let's
name it set e another way of writing
that is using the symbol the null set
the cardinality of set e is equal to
zero this notation simply means the
number of elements singleton set assert
with one element set s is a singleton
set the cardinality of set s is one
finite set a set with limited elements
example set F is a set of counting
numbers less than 6 the elements of set
F are 1 2 3 4 5 the cardinality of set F
is 5 infinite set a set width and
limited elements example set n is the
set of counting numbers the element
subset n are 1 2 3 and so on the
cardinality of set n is infinite equal
sets two sets are equal if they have the
same elements example set E is a set of
letters in the word earth the element
subset e r e a R th set H is a set of
letters in the word heart the elements
of set H are H E a R T set E and set H
have the same elements they are equal
sets set e equals set H
equivalent sets equivalent sets have
different elements but have the same
number of elements example given set F
and set H the cardinality of set F is 5
and the cardinality of set H is 5 the to
sets have different elements but they
have the same number of elements set F
is equivalent to set H Universal set
Universal set is the set containing all
elements at which all other sets are
subsets the universal set is represented
by a capital letter you example let u be
the universal set set u is the set of
all flowers in the English alphabet all
other sets are subsets of the universal
set set a is a proper subset of set you
set B is also a proper subset of set u
all the elements of sets a and B are
also found in the universal set another
example set U is a set of counting
numbers sets a B and C are proper
subsets of set u because a universal set
contains all the elements in a
particular problem subset set a is a
subset of set D if and only if every
element in a is also an element in B
here's an example
list of the possible subsets upset a
number-one asset with three elements two
three four sets with one element five
six seven sets with two elements in
number eight is an empty set how many
subsets does set a health to find the
number of subsets use this formula two
to the power of n where n is a number of
elements the cardinality of set a is 3
so 2 to the power of 3 is just 2 times 2
times 2 equals 8 proper subset set a is
a proper subset of set B if there is at
least one element in B not contained in
a in symbol set a is a proper subset of
set B example given sets a B C and D set
B is a proper subset of set a set C is a
proper subset of set a set D is a proper
subset of set a list all the proper
subsets of set a set one to three have
two elements sets four five six have one
element and number seven is an empty set
the set itself is not a proper subset of
set a how many proper subsets does set a
half since the set itself is not a
proper subset the formula is 2 to the
power of n minus 1 where n is the number
of elements the cardinality of set a is
3 using the formula to ^
3 minus 1 equals 8 minus 1 is 7
supper set supper set is a set
containing all of the elements of
another set symbol set a is a supper set
of set B this is just a reverse of
subset proper superset set a is a proper
supper set of set B if set a is a
superset of set B and set a is not equal
to set B in symbol set a is a proper
superset of set B given set a and set B
well set B is a proper subset of set a
set a is a proper superset of set B
power set the set of all the subsets of
a set what is the power set of set s EO
Bess contains all the subsets of set s
how many elements does the power set of
set s have since the power set contains
all the subsets of a given set it
follows the formula two to the power of
n where n is the number of elements so
the number of elements of the OBS is two
to the power of three which is equal to
eight
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