Selected items from the Introduction to set theory and to methodology and philosophy of mathematics and computer programming.

1. 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

2. 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.

3. 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.

4. 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?

5. 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.

6. 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.

7. 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) and
denote 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, yi
where x ⌘3 y and x, y 2 { 2..6}.