Equality

Date: 2025-11-22
"=" is an indicator function over a set of ordered pairs that checks if first and second elements are the same

So for the set {1,2,3}, it is defined as  =:{1,2,3}×{1,2,3}->{0,1}

= (1,2) => 0 (not equal)
= (3,3) => 1 (equal)

There are 2 types of equality.

Let the set X = {a, 3, 6, v} consist of 4 keyboard symbols.

The notation a = a means that a is equal to itself, i.e.,it is one and the same object.
a = a if they cannot be distinguished by applying the same functions to them.

Let a function d: X → X be such that:
a ⇾ 6
6 ⇾ 3
3 ⇾ 3
v ⇾ a

Let's compute d 3:
d 3 => 3
That is, we can write:
d 3 = 3 — if we evaluate the expression d 3, we get 3.

d 6 = 3 — since d 6 => 3.

The second type is equivalence (belonging to the same class).

Let e be an indicator function e: X → {digit, letter}, defined intuitively as:
a ⇾ letter
6 ⇾ digit
3 ⇾ digit
v ⇾ letter

From the function e, we can form two sets — {a, v} and {6, 3}.

Example:
e v = e a

Let's compute to verify the equality:
e v => letter
e a => letter

The diagram proves the equality. The arrows show the transformation (application of the function).
 e v   =  e a
  ↓        ↓
letter = letter
Equality

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