What is the difference between Logical Implication and Logical Equivalence?

Question

I am studying mathematical logic and programming logic, and I would like to know the difference between Logical Implication and Logical Equivalence?

Answer 1

Logical implication

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

p → q

And she 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 logical equivalence that follows:

Logical equivalence

p ↔ q

Which means:

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

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

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 by simplifying:

p = q

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

to = 0 b = 0 ab = 0

In relation to implication we have the following:

to → b
b → to
----
∴ to ↔ b

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

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

to ↔ b
----
∴ to → b ∧ b → to

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 exclusive or-exclusive means this:

p ≠ q

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

Answer 2

The "Logical Implication" is established between two concepts or propositions, in such a way that the affirmation of the truth of one leads to the necessary inference of the truth of the other. It indicates that one condition must necessarily be satisfied for the other to be true.

Ex: if...else

The "Logical Equivalence" is a relation of logical equality or mutual implication between two propositions, such that each of them is true only if the other is true.

Ex: x===y