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Functions and Relations (CSCI 2824, Spring 2015)

In this section, we learn the notion of relations, functions and well-definedness.

This is part of Sect 4.1 of the book.

Functions

We have already encountered functions many times and we often use functions when we program. Let us formalize what functions are in mathematical terms:

Functions

A function f from a set A to a set B is associates (or maps) every element of the set A to some element of the set B. We express the fact that a function f maps A to B as f: A rightarrow B.

Example -1

Consider A = { 1,3,5,7,11 } and B = { 2,4,6,8,12 }. Consider function f: A rightarrow B.

 

The function above can be represented explicitly by the mapping

1 mapsto 2, 3 mapsto 4, 5 mapsto 6, ldots

or implicitly by the rule/formula f(x) = x+1.

Example-2

Functions can map many elements of A to the same element of B. But the rule is that each element of A can only be mapped to one element of B. Which of the following are functions?

 

Example-3: Power Set Signature

While discussing power sets, we mentioned a correspondence between power-sets of a set and binary strings. We can indeed write it as a function. Let us take A to be {1,2,3} and B_3 = { 000,001,010,011,101,110,111 }. We write the function s: P(A) mapsto B_3 by using the mapping from every element of the power set S in P(A) and every element of B_3.

The table below shows the elements of the power set and the corresponding value of s(x).

Power set elt. (x in P(A)) s(x)
{} 000
{a} 100
{b} 010
{c} 001
{a,b} 110
{a,c} 101
{b,c} 011
{a,b,c} 111

Example-4: Mathematical functions

You may have seen many examples of functions from your mathematics classes, thus far.

  • k(x): x+1, a function from mathbb{N} rightarrow mathbb{N} takes an input natural number and yields an output natural number.

  • f(x): sin(x) is a function from mathbb{R} rightarrow mathbb{R}, takes an input real number and yields a real number output.

  • g(x): e^x is a function from mathbb{R} rightarrow mathbb{R}, takes an input real number and yields a real number output.

  • h(z): lfloor z rfloor is a function from mathbb{R} rightarrow mathbb{Z}, , takes an input real number and yields an integer output.

Non-Examples of Functions.

It remains to clarify what is not a function. In general, a mapping from set A to set B can fail to be a function for either of the two reasons below:

  1. It leaves an element of set A unmapped.

  2. It maps an element of A to multiple elements in B.

If either case occurs then, the mapping fails to be a function.

Consider the two mappings shown below. The mapping on the left fails to be a function because it does not map the elements 1,2 from the domain, while the mapping on the right fails to be a function since 2 is mapped to multiple elements.

 

Mathematical non-functions

Strictly speaking, many functions that we saw in calculus are not quite functions.

  • The function frac{1}{x} is not a function from mathbb{R} rightarrow mathbb{R}.

    • It is a function however from (mathbb{R} - {0}) rightarrow mathbb{R}. Here, we have removed the input x=0 from the domain.

  • The function log(x) is not a function from mathbb{R} rightarrow mathbb{R} since it leaves all values x leq 0 unmapped.

    • If we wish to be precise, we will define the type of log(x) to be mathbb{R}_{geq 0} rightarrow mathbb{R}.

  • The function sqrt{x} is not defined for x < 0 and can be defined to both positive and negative square roots when x geq 0. For instance sqrt{4} = pm 2.

    • If we wish to be precise, we define | sqrt{x}| to be a function from mathbb{R}_{geq 0} rightarrow mathbb{R}, with the domain restricted to non-negative reals.

Relations

Formally, a relation R between sets A and B is defined as a subset of A times B, i.e, R subseteq A times B.

Example-1

Let us take A = { 1,2,3,4,5,6} and B = {1,2,3,4,5,6}. Consider the relation R subseteq A times B by

R = { (1,1), (2,1), (3,1), (4,1), (5,1), (6,1), (6,2), (6,3) }

Like functions, we may view the relation as a mapping. However, unlike functions, it is possible that

  • x in A is not related to any element in B.

  • x in A could be related to multiple elements in B.

The relation R is visualized below:

 

Functions are a special case of relations, wherein

  • xin A is mapped to exactly one element in B.

Example-2

Here is another example of a relation over numbers: R subseteq mathbb{N} times mathbb{N} defined as

R = { (m,n) | m mbox{divides} n }

Write down some examples of elements of R.

Example-3

Consider sets A = {1,3,5} and B = {2,4,6} write the function f: A rightarrow B defined by f(n) = n+1 as a relation R_f subseteq A times B.

Counting Relations

If A is a set with m elements and B with n elements then how many relations can exist between A and B? How many functions?

Answer in class.

Domains/Co-Domains

Let f be a function f: A rightarrow B. We say that A is the domain of f and B is the co-domain.

Similarly, let R subseteq A times B be a relation. We say that A is the domain of the relation and B is the co-domain.

Relations On a Set

A relation R from set A to itself is called a relation on A . We can represent relations from a set to itself by a special diagram called a graph.

E.g., Let A = { 1,3,5,7,9 }. Consider a relation R = { (a,b) | a=b lor b = a+2 }.

The relation has the tuples R = { (1,1), (1,3), (3,3), (3,5), (5,5), (5,7), (7,7), (7,9), (9,9) }. The graph looks as follows:

 

The rule is we have nodes or vertices for each element in the set A. If (a,b) in R, we draw an arrow from a to b. Graphs are very useful as visualizations of relations. We will spend 2-3 weeks at the end of this course talking about properties of graphs.