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)

In complex expressions where "=" is written between elements, this refers to the notation where the operator is placed between operands 
it is customary to evaluate it at the end, after the other functions have been evaluated in order to correctly check final equality

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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