Foundational Knowledge for Functions

What is a Function?

A function involves the following: 2 sets, elements within the respective sets, & the relationship between the 2 sets.

Let’s call the 2 sets — Set A & Set B. For the function to be a function, all the elements in Set A has to have at most one relationship with some element in Set B.

This relationship is commonly known as mapping, and symbolically looks something like this:

𝑓: A→B

• Capital letter “A” represents Set A, whilst letter “B” represents Set B.
• “𝑓” is only considered a function if “𝑓”assigns each element a(let’s represent each element in Set A with a small letter “a”) within Set A to one element in Set B

Example:

𝑓: 𝑥 →𝑥²

• 𝑥 represents Set A, whilst 𝑥² represents Set B
• 𝑥 undergoes a transformation via the function 𝑓 to become 𝑥²
• If 𝑥=2 (an element within Set A), the function 𝑓 maps 𝑥=2 to an element within Set B, which we know to be 4.

Terminology:

• Set A is known as the domain
• Set B is known as the co-domain
• Elements within Set B is known as the range or image
• A “function” that does not map all elements from Set A to Set B is not a function but a “partial function”

Different Types of Functions

The following descriptions are in my own words for quick but essential understanding for the 3 types of functions in basic discrete math.

Injective — Every unique element in the domain maps to at most one other unique element in co-domain, hence, also known as a “one-to-one” function.

Alternatively, if more than one element in the co-domain is being mapped to more than once by any element in the domain, it is no more injective.

Surjective — Every element in the co-domain is mapped onto by some element from the domain, also known as “onto” function.

Alternatively, if there is any element in the co-domain that is not being mapped to, it is no longer surjective.

Bijective — Every unique element in the domain maps to at most one other unique element in co-domain AND every element in the co-domain is being mapped onto. It is both injective & surjective at the same time.

Alternatively, if any element in the co-domain is not being mapped onto OR if any element in the domain points to more than one element in the co-domain it is not longer bijective.

Even more simply put, a bijective function has to have every element be mapped to/from only once.