Properties of Injective and Surjective Functions

The video tutorial explains the concepts of one-to-one (injective) and onto (surjective) functions, providing examples and mappings to illustrate these ideas. It also covers bijective functions, which are both injective and surjective, and includes practice problems to help viewers identify different types of functions. The tutorial concludes with an analysis of more complex functions, emphasizing the importance of understanding these foundational concepts in mathematics.

What is the main characteristic of an injective function?

• Each element in the range is mapped to by multiple elements in the domain.

• Each element in the domain maps to at least one element in the range.

• Each element in the range is mapped to by exactly one element in the domain.

• Each element in the domain maps to multiple elements in the range.

In an injective function, if f(a) = f(b), what can we conclude about a and b?

• a and b must be in the range.

• a and b can be any values.

• a and b must be equal.

• a and b must be different.

What happens if an element in the range of an injective function maps to two elements in the domain?

• The function becomes surjective.

• The function remains injective.

• The function becomes bijective.

• The function is no longer injective.

What is a key feature of a surjective function?

• Every element in the range is mapped to by multiple elements in the domain.

• Every element in the domain maps to multiple elements in the co-domain.

• Every element in the domain maps to a unique element in the range.

• Every element in the co-domain is mapped to by at least one element in the domain.

In a surjective function, what must be true about the co-domain?

• It must be equal to the range.

• It must be smaller than the domain.

• It must contain elements not mapped to by the domain.

• It must be larger than the domain.

What defines a bijective function?

• It is both injective and surjective.

• It is only surjective.

• It is neither injective nor surjective.

• It is only injective.

If a function is bijective, what can be said about its domain and co-domain?

• The domain is larger than the co-domain.

• The co-domain is larger than the domain.

• Each element in the domain maps to a unique element in the co-domain, and vice versa.

• The domain and co-domain are unrelated.

In the practice problem, why is the function not one-to-one?

• Because the co-domain is not fully utilized.

• Because the range is larger than the domain.

• Because each element in the domain maps to multiple elements in the range.

• Because each element in the range is mapped to by multiple elements in the domain.

Why is the function in the practice problem not onto?

• Because not every element in the co-domain is mapped to by the domain.

• Because the domain is larger than the co-domain.

• Because the function is injective.

• Because the range is equal to the co-domain.

For the function f(x) = x^2, why is it not one-to-one?

• Because it maps multiple elements in the domain to a single element in the range.

• Because f(a) = f(b) does not imply a = b.

• Because it maps a single element in the domain to multiple elements in the range.

• Because f(a) = f(b) implies a = b.