Lecture 06 Part 2

Relations

Can you name all of the following symbols?

These symbols are called relations. Their job is to establish a connection between two numbers. When any of these symbols are found between two numbers, it means that the two numbers satisfy a certain condition required for such relation. For example, when we write

it means that 1.999 and 2 satisfy the relation of being "approximately equal" to each other. Note that relations are different from operations like plus (+), minus (-), times (×), and divide (÷). Operations generate a certain result while operations are only statements which may be identified as either true or false.

In this part of Lecture 06, we will understand what relations are in the theoretical level and how they are used to aid mathematical work.

Cartesian Product

Given two non-empty sets A and B, their Cartesian product A × B is the set of all ordered pairs (a,b) where a ∈ A and b ∈ B. Mathematically,

For example, if A = {1, 2, 3} and B = {1, 2}, then

Question. Is A × B = B × A?

It is also possible to have a Cartesian product of a set with itself. For example,

For faster writing we sometimes denote these by

In some cases, triple (and even multiple) products are defined like A × B × C, and

However for this class, we will not be dealing with multiple Cartesian products. Instead we will restrict our coverage with pairs.

Exercise. Given A = {a, b}, B = {1, 2, 3}, find

Exercise. If A has 3 elements and B has 2 elements, how many elements does A × B have? How about B × A? What about A² and B²? In general, if A has n elements and B has m elements, how many elements does A × B have?

Cartesian products are used in mathematics to construct a set out of other sets that are useful for some purpose. For example, the Cartesian plane which is a set of ordered pairs (x, y) where x and y are both real numbers is simply the Cartesian product

In probability, Cartesian products express the sample space of experiments. For example, if a man has two t-shirts of colors S = {red, green} and two pants of colors P = {black, blue}. Then all the possible outfits he could wear are given by the Cartesian product:

S × P = {(red, black), (red, blue), (green, black), (green, blue)}

Relations

Given a set A, we define a relation on A as a subset of A². For now, we will denote relations by R. For example, if A = {1, 2, 3}, then A² = {(1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3)}. Thus, any of the following can be considered as relations on A.

If R is a relation on A and (a,b) ∈ R, then we can write aRb. Hence, from above, we can say

Exercise. From the example above, determine whether the following are true or false.

Exercise. Let A = {a, b, c}. Let ~ (called "tilde", read as "TIL-duh") be a relation on A defined by

~ = {(a,a), (a,b), (a,c), (b,a), (b,b), (b,c), (c,c)}

Determine whether the following are true or false.

From this point forward, we will mostly use symbols like ~ for relations instead of the letter R.

Reflexive, Symmetric, and Transitive Relations

Some relations are more important than others because they satisfy certain special properties. Some kinds of relations that we are interested in are reflexive, symmetric, and transitive relations.