Suppose that f is a function from A to B, where A and B are finite sets with |A|=|B| . Show that

Suppose that f is a function from A to B, where A and B are...

Key Concepts

Injective Function

An injective function, also known as a one-to-one function, is one in which distinct elements in the domain map to distinct elements in the codomain. In other words, if f(x) equals f(y), then x must equal y. This property is crucial for ensuring that no two different inputs produce the same output, laying the foundation for establishing a correspondence between sets.

Surjective Function

A surjective function, or onto function, is defined by the property that every element in the codomain is the image of at least one element from the domain. This means the function covers the entire codomain. Surjectivity is key in ensuring that there are no 'unused' elements in the codomain, thereby creating a complete mapping from the domain to the codomain.

Bijective Function

A bijective function is one that is both injective and surjective. In such a function, every element of the domain maps to a unique element in the codomain, and every element of the codomain is mapped by exactly one element of the domain. Bijectivity signifies a perfect one-to-one correspondence between sets, which is particularly important when the two sets have the same finite cardinality.

Finite Sets and Cardinality

Finite sets are sets with a countable number of elements, and the concept of cardinality refers to the number of elements within a set. When working with functions between finite sets of equal cardinality, the structure of the sets forces injectivity and surjectivity to imply one another, which is a key observation in establishing bijective relationships.

Pigeonhole Principle

The pigeonhole principle is a fundamental counting argument which states that if more objects (pigeons) are distributed into fewer boxes (pigeonholes) than there are objects, then at least one box must contain more than one object. In the context of functions between finite sets with equal cardinality, this principle underpins the reasoning that a function cannot be injective without also being surjective, and vice versa, because any deviation would contradict the available number of elements.

Transcript

00:01 For this problem, we have been told that we have two sets, a and b, and these are finite sets.

00:06 In other words, there is a fixed number of elements within each set.

00:11 And the cardinality of the two sets are equal.

00:13 So the number of elements in set a is equal to the number of elements in set b.

00:18 We're also told we have a function going from a to b.

00:21 What we want to show in this problem is that f is one to one, if and only if it is onto.

00:28 So we're going to go both ways.

00:31 We're going to assume that f is one to one and show that that means it has to be onto.

00:36 And then we'll assume that f is onto and show that it has to be one to one.

00:40 Now before we do that, though, let's do some definitions.

00:45 What does it mean if a function is one to one? well, a function is said to be one to one if and only if, if we have f of a equaling f of b, that implies that a equals b.

01:01 In other words, every a only maps to one b.

01:05 That's a definition of a function.

01:07 But every b only comes from one a.

01:12 So it goes both directions.

01:15 Each one, every a points to a b.

01:17 Every b comes back to a single a.

01:19 So that's one to one.

01:23 What about onto? a function is onto if for every element b within that, set b.

01:34 So every element in that b, there is an element a within the first set, such that f of a equals b.

01:43 In other words, every single element in that second set b is mapped to an element in set a.

01:50 So those are our two definitions.

01:53 So let's take the first piece.