Understanding Functions: Injective, Surjective, and Bijective

The video tutorial explains the concepts of injective, surjective, and bijective functions using a set with elements one, two, three, and four as both domain and codomain. It reviews the definitions of these functions and analyzes four specific functions to determine their properties. Function 1 is neither injective nor surjective, Function 2 is bijective, Function 3 is also bijective, and Function 4 is neither injective nor surjective. The tutorial uses two-line notation to simplify the analysis of functions.

What is a characteristic of an injective function?

• The function misses some elements of the codomain.

• Every element of the domain is mapped to a unique element in the codomain.

• No element of the codomain is mapped by more than one element of the domain.

• Every element of the codomain is mapped by at least one element of the domain.

Why is the first function not surjective?

• It has repeated elements in the codomain.

• It misses some elements from the codomain in the range.

• It maps multiple elements of the domain to the same element in the codomain.

• It maps each element of the domain to a unique element in the codomain.

What makes the second function bijective?

• It is both injective and surjective.

• It is only injective.

• It is only surjective.

• It is neither injective nor surjective.

In the second function, what ensures it is injective?

• All elements of the codomain appear in the range.

• No element of the codomain is repeated in the range.

• Some elements of the codomain are missing in the range.

• The function maps multiple elements of the domain to the same element in the codomain.

What is the significance of using two-line notation for functions?

• It makes it easier to determine the type of function.

• It complicates the understanding of the function.

• It is only useful for injective functions.

• It is only useful for surjective functions.

Why is the third function considered bijective?

• It misses some elements from the codomain.

• It has repeated elements in the codomain.

• Each element in the codomain is mapped by exactly one element from the domain.

• It maps multiple elements of the domain to the same element in the codomain.

What is the result of the fourth function when x is even?

• x minus one divided by two

• x plus one divided by two

• x divided by two

• x multiplied by two

Why is the fourth function not injective?

• It misses some elements from the codomain in the range.

• It maps each element of the domain to a unique element in the codomain.

• It has repeated elements from the codomain in the range.

• It maps multiple elements of the domain to the same element in the codomain.

What is a characteristic of a surjective function?

• Every element of the codomain is mapped by at least one element of the domain.

• It misses some elements from the codomain in the range.

• No element of the codomain is mapped by more than one element of the domain.

• It maps each element of the domain to a unique element in the codomain.

What is the outcome when x is odd in the fourth function?

• x plus one divided by two

• x divided by two

• x minus one divided by two

• x multiplied by two