Introduction to set theory and to methodology and philosophy of

mathematics and computer programming

Basic binary relations

An overview

by Jan Plaza

c�2017 Jan Plaza

Use under the Creative Commons Attribution 4.0 International License

Version of March 25, 2017

Definition

Let X,Y be any sets.

1. The full binary relation on X , denoted 1X2 , is X ⇥X.

2. The full (binary) relation on X,Y , denoted 1X,Y , is X ⇥ Y .

3. The empty relation is ;.

4. The equality relation on X , denoted =X , is {hx, xi 2 X ⇥X : x 2 X}.

5. The disequality relation on X , denoted 6=X , is {hx, yi 2 X ⇥X : ¬(x=y)}.

The term“inequality” is reserved for 6, >, <, >.

Exercise. Let X = {1, 2, 3, 4}. Make discrete Cartesian graphs of the relations above.

Definition

Let X ✓ R.1. The less-than-or-equal/smaller-than-or-equal relation on X ,

denoted 6X , is {ha, bi 2 X ⇥X : a 6 b}.

2. The greater-than-or-equal relation/bigger-than-or-equal on X ,

denoted >X , is {ha, bi 2 X ⇥X : a > b}.

3. The less-than/smaller-than relation on X ,

denoted <X , is {ha, bi 2 X ⇥X : a < b}.

4. The greater-than/bigger-than relation on X ,

denoted >X , is {ha, bi 2 X ⇥X : a > b}.

Exercise. Let X = {1, 2, 3, 4}. Make discrete Cartesian graphs of the relations above.

Definition

Let X be any family of sets.

1. The subset/inclusion relation on X , denoted ✓X , is

{ha, bi 2 X ⇥X : a ✓ b}.2. The superset relation on X , denoted ◆X ,

is {ha, bi 2 X ⇥X : a ◆ b}.3. The proper subset/inclusion relation on X , denoted ⇢X ,

is {ha, bi 2 X ⇥X : a ⇢ b}.4. The proper superset relation on X , denoted �X ,

is {ha, bi 2 X ⇥X : a � b}.

Exercise. Let X = P({1, 2}). Why is {1} ✓X {1, 3} false?

In ZFC, a relation must be a set (of ordered pairs).

6 is the same as 6R , and it is a relation because it is a subset of R⇥ R.

= is not a relation, because {hx1, x2i : x1=x2} is not a set.

✓ is not a relation, because {hx1, x2i : x1 ✓ x2} is not a set.

2 is not a relation, because {hx1, x2i : x1 2 x2} is not a set.

(They might be relations in the sense of a metatheory of ZFC.)

=X is a relation because {hx1, x2i 2 X ⇥X : x1=x2} is a set.

✓X is a relation because {hx1, x2i 2 X ⇥X : x1 ✓ x2} is a set.

Definition

By the divisibility relation on Z , denoted by the mid symbol | , we understand

the binary relation consisting of all the ordered pairs hm,ni 2 Z ⇥ Z such that

9k2Z n = k ·m. The formula m|n is read “m divides n”.

Notes

According to the definition above, “m divides n” is equivalent to

“n is a multiple of m”, in all cases, even if m = n = 0.According to the definition above, 0|0.This does not mean that we allow the division 0/0.

Definition

Let k 2 Z and k > 1. By the congruence modulo k we understand the binary

relation consisting of all the ordered pairs hm,ni 2 Z ⇥ Z such that k|(n�m) anddenote it by ⌘k . The formula m ⌘k n is read “m is congruent to n modulo k”.

Instead of m ⌘k n one can write m ⌘ n(mod k) .

1. In the definition of congruence mod k the condition k|(n�m) can be replaced by

k|(m� n) – the two conditions are equivalent.

2. Congruence modulo k is of importance to cryptography, which in turn is of

importance to the Internet and computer network security.

Exercise

Make a discrete Cartesian graph of ⌘3 which shows only those ordered pairs hx, yiwhere x ⌘3 y and x, y 2 {�2..6}.