Logical implication

Starting from my answer in your other question . We have that a logical implication is a clause in the following Form:

p → q

And it means:

If p is true, then q is also true.

But what if p is false? Well, in this case, we can say nothing about q without further information. That is, q May or may not be true.

This contrasts with the following logical equivalence:

Logical equivalence

p ↔ q

Which means:

p is true if, and only if, q is also true.

In this case, if p is false, then q is also false, after all q can only be true if p so is it.

This is called equivalence because of the following:

If p is true, q is also true.
If p is false, q is also false.

Or simply put:

p = q

It happens that the equality in logic is called the equivalence. For equality, the =, while for the equivalence, the ↔. This might seem silly in principle, but it serves so that logical expressions with mathematical subexpresss like the following can be read accurately without needing too many parentheses:

a = 0 ∨ b = 0 ↔ ab = 0

Regarding implication we have the following:

to → b
b → the
----
∴ a ↔ b

And that is why the symbol is used ↔, for logical equivalence can also be interpreted as a two-way implication.

And obviously the demonstration can also be done in the opposite direction:

a ↔ b
----
{ a → b ∧ b → a

Or-exclusive

Finally, we still have the or-exclusive, which is the exact opposite of equivalence. One way to present it is like this:

Or p is true, or q is true, but not both.

Now, in this case, it is not difficult to conclude that this means that if only one of them can be true, then when one is true, the other will automatically be false, and when one is false, the other will automatically be true. And therefore, the or-unique means this:

p ≠ q

But again, so we don't need too many parentheses in logical expressions with mathematical subexpressions, some other symbol is used to denote the unique or -. The most common ones I see out there are ⊻ e ⊗, although there are others.

The "logical implication" is established between two concepts or propositions, such that the affirmation of the truth of one of them leads to the necessary inference of the truthfulness of the other. It indicates that one condition must necessarily be satisfied for the other to be true.

Ex: if...else

"logical equivalence" is a relationship of logical equality or mutual implication between two propositions, in such a way that each of them is true only if the other also for.

Ex: x===y