What are relations?

In every day life, a person is related to their parents, siblings, cousins, teachers, friends, etc. in some way. Similarly in mathematics, mathematical objects like numbers and sets are related to one another in some way. Many relations (symbols) will be familiar already:

1. $2 <3$
2. $\pi \approx 3.14$
3. $5 \in \mathbb{Z}$
4. $X \subset Y$
5. $a \equiv b(modn)$

Consider the following set $A = \{1, 34, 56, 78 \}.$ We can compare the numbers in A using the symbol “<” as follows: $1 < 56, 34 < 78$ etc. We can write this as a set in the following way: $R = \{ (1,34), (1,56), (1,78), (34,56), (34,78), (56,78) \}.$

Each pair in this new set R expresses the relationship x < y (where x and y are numbers from A).

In other words, $1<34, 1<56, 1<78, ...$ So if asked whether $34 < 78$ is true,  one only needs to look into our set R to find the pair $(34,78).$ If we didn’t find it, then the relation would be considered false for the given set. The above example is intuitive because we are already comfortable with the relation <. In more abstract cases, thinking of the relationship between mathematical objects in this way may be a little trickier!

The formal definition for a relation is: A relation on a set A is a subset R ⊆ A × A. We often abbreviate the statement (x, y) ∈ R as xRy.

Note that this tells us that the relation R is a subset of the Cartesian Product A × A. Hence it is an explicit summary of the relations between objects in set A.

Consider the next example:

$A =\{1, 50, 100, 400 \}$ is the set.

$R = \{ (1, 50), (1,400), (50,100),(100,400) \} \subseteq A \times A$ is the relation.

Then we have 1R50, 1R400, 50R100 and 100R400 – where R is the relation between elements in set A. We also know 1 and 100 are not related, and similarly 50 and 400 are not related, hence not in set R.

Note, $A \times A = \{ (1,1), (1,50), (1,100), (1,400), (50,50), (50,100), (50,400), (100,100), (100,400), (400,400) \}$ is the Cartesian Product of set A.

Consider the following example: $R = \{ (x,y) \in \mathbb{Z} \times \mathbb{Z} : x - y \in \mathbb{N} \} \subseteq \mathbb{Z} \times \mathbb{Z}.$ This is the > relation on the set $A = \mathbb{Z}.$ R is infinite, but here are some values of R: $R = \{(-2,-3), (-1, -8),..., (100,50), (100,51) ...\}.$ The value of x must always be larger than the value of y.

 How clear is this post?