RELATIONS - DISCRETE MATHEMATICS

Summary

In this video, the concept of relations is introduced with a clear explanation of its definition. The presenter defines a relation R on a set X as a subset of X * X, where two elements from the set X are involved. The notation ‘x R y’ is explained to show that if an element (a, b) is part of the relation, it can be written as ‘x R y.’ The session aims to demystify relations with simple examples, ensuring that the audience grasps the concept easily and thoroughly.

Takeaways

• 😀 A relation R on a set X is a subset of X * X.
• 😀 A relation contains pairs of elements (a, b) from the set X.
• 😀 The notation 'x R y' is used to represent that (x, y) belongs to the relation R.
• 😀 The concept of relations can seem more complicated than it actually is.
• 😀 The focus will be on understanding relations through examples.
• 😀 A relation is a mapping from one element to another within a set.
• 😀 The elements of a relation are ordered pairs (a, b) of elements from X.
• 😀 The goal is to break down the concept of relations into easily understandable steps.
• 😀 By looking at examples, the idea of relations becomes much clearer.
• 😀 The aim is to make learning about relations simple and straightforward.

Q & A

What is a relation R on a set X?

A relation R on a set X is a subset of X * X, which means it consists of ordered pairs (a, b) where both a and b are elements of X.

How is a relation between two elements of set X written?

If (a, b) is in the relation R, it is written as 'a R b'.

What does the notation X * X represent in the context of relations?

X * X represents the Cartesian product of set X with itself, which consists of all ordered pairs (a, b) where both a and b are elements of X.

Why is understanding relations considered easy in this explanation?

The explanation emphasizes that although the concept may seem complex at first, the actual application of relations is very straightforward and can be easily understood with examples.

What role do examples play in understanding relations?

Examples are used to demonstrate how relations work in practice, helping to simplify the concept and make it more relatable.

What does it mean for (a, b) to be in a relation R?

For (a, b) to be in a relation R, it means that there is a defined connection between the elements a and b within the set X, and they are part of the subset forming the relation.

How does the relation R help in understanding the connections between elements?

The relation R provides a formal way of describing which elements are connected to each other in a set, offering insights into the structure and interactions within the set.

Is the concept of a relation only applicable to sets with two elements?

No, relations are applicable to any set X and can involve any number of elements, as long as the relation is defined as a subset of X * X, which includes all possible pairs of elements.

Can a relation have only one element in the set X?

Yes, a relation can be defined on a set with only one element, but in that case, the only possible pair would be (a, a), where a is the single element of the set.

What makes a relation a 'subset' of X * X?

A relation is a subset of X * X because it contains only some of the ordered pairs from the Cartesian product X * X, not necessarily all of them.