4.1. Definition

Today we're going to start with the definition of a relation. Can anyone tell me what a relation actually is?

Isn't it something to do with how two things connect?

Exactly! A relation between two sets A and B is a subset of the Cartesian product A×B. In simpler words, it consists of ordered pairs where the first element is from A and the second is from B. For example, if we have A = {1, 2, 3} and B = {a, b, c}, one possible relation could be R = {(1, a), (2, b), (3, c)}.

Wait, so it’s like matching pairs?

Yes! You can think of it as matching pairs from two different groups. Now, can anyone tell me what a Cartesian product is?

Isn't that when you combine all elements from both sets in ordered pairs?

Spot on! The Cartesian product A×B creates all possible pairs from the two sets. Let’s move on to discuss different types of relations.

Now that we understand what a relation is, let’s break down the types of relations. First, we have the reflexive relation. Can anyone guess what that means?

Does it involve pairs like (x, x)?

Exactly! A relation R is reflexive if every element x in set A also has the pair (x, x) in R. For example, if A = {1, 2, 3}, then the reflexive relation could be R = {(1, 1), (2, 2), (3, 3)}. What about symmetric relations?

That sounds like if (a, b) is in R, then (b, a) is also in R?

Precisely! Next, we have transitive relations. Can anyone explain what that means?

If you have (a, b) in R and (b, c) in R, then (a, c) should also be in R?

Yes, wonderful! And remember, an anti-symmetric relation happens when if (a, b) and (b, a) are both present, then a must equal b. Learning these types helps in identifying how relations behave.

Let's switch gears to functions. Can anyone tell me what makes a function different from a relation?

Functions map inputs to one specific output, right?

Exactly! A function is a special kind of relation where each input from the domain maps to exactly one output in the co-domain. We denote this as f: A → B. For instance, if A = {1, 2, 3} and B = {a, b, c}, then a function could look like f = {(1, a), (2, b), (3, c)}.

What about injective or surjective? I’ve heard those terms.

Great question! An injective function means different inputs yield different outputs. Surjective, or onto, means every element in the co-domain is mapped by at least one element in the domain. And if a function is both, we call it bijective. Let’s take an example: if f = {(1, a), (2, b), (3, b)}, it’s surjective because both 2 and 3 map to b, while f = {(1, a), (2, b), (3, c)} is bijective. This is crucial as we delve deeper into functions later.

Now let's discuss the composition of functions. If we have two functions f and g, how do we define their composition?

Isn’t it like applying one function after another? Like, g(f(x))?

Correct! The composition of f and g, denoted g∘f, is defined as g(f(x)) for all x in A. This means you apply function f first, then use the result in function g. Can anyone tell me what the inverse of a function is?

Does it reverse the operation of f?

Exactly! The inverse f⁻¹ undoes what f does. A function has an inverse only if it is bijective. So, if we were to say f: A → B, the inverse would be f⁻¹: B → A. This concept is vital to understand how functions relate to each other.