Numbers have properties and we use these properties to define sets of numbers.

• Evens: \(\{ \cdots , -6, -4, -2, 0, 2, 4, 6, \cdots \} = \{ 2n \, | \, n \in \mathbb {Z} \}\)
• Squares: \(\{ 1, 4, 9, 16, \cdots \} = \{ n^2 \, | \, n \in \mathbb {N} \}\)
• Primes: \(\{ 2, 3, 5, 7, 11, \cdots \}\)

Once we know that a number belongs to one of these sets, i.e. has the property, then we automatically know some information about the number.

\(\blacktriangleright \) Same with functions.

A major goal of this course is to develop a list of function properties and define sets of functions via these properties. Once we know a function belongs to a set, then we get a lot of free information about the function.

Basic Function Characteristics or Properties

Let’s start off with two basic properties of functions: Onto and One-to-One.

Some functions are onto. Some are not.

The one and only rule for a function is that each domain item is in exactly one pair. Range items can be in multiple pairs or just one pair.

However, if the range also follows this rule, then the function is called a one-to-one function.

Some functions are one-to-one. Some are not.

\(\blacktriangleright \) Example: A function that is both onto and one-to-one.

(arrows go from the domain to the codomain.)

\(\blacktriangleright \) Example: A function that is onto but not one-to-one.

(arrows go from the domain to the codomain.)

\(\blacktriangleright \) Example: A function that is one-to-one but not onto.

(arrows go from the domain to the codomain.)

\(\blacktriangleright \) Example: A function that is neither one-to-one nor onto.

(arrows go from the domain to the codomain.)

These examples illustrate that onto and one-to-one are independent properties. A function can have either property with or without the other.