# Logical equivalence

Logical equivalence is a relationship between two or more formulas that have the same truth value in every possible interpretation. In other words, the formulas are logically equivalent if they always produce the same result, regardless of the values of the variables involved.

There are a few different ways to show that two formulas are logically equivalent. One way is to use a truth table, which shows the truth values of the formulas for all possible combinations of truth values for the variables involved. Another way is to use a proof by contradiction, in which it is shown that if the formulas were not logically equivalent, then that would lead to a contradiction. What do you mean by logical equivalence? Logical equivalence is a relationship between two statements that are either both true or both false. In other words, the statements are logically equivalent if they have the same truth value. What is logically equivalent to P → Q? There are a few different ways to answer this question, but one way to think about it is that P → Q is logically equivalent to ¬P ∨ Q. In other words, if P is true, then Q must also be true. If P is false, then Q can be either true or false.

How do you determine logical equivalence? To determine logical equivalence, you need to show that two formulas are equivalent in all possible truth assignments. This can be done by constructing a truth table for each formula and comparing the results. If the formulas produce the same results in every possible truth assignment, then they are logically equivalent.

Which is logically equivalent to P ∧ Q → R? There are a few different ways to answer this question, but one way to think about it is that P ∧ Q → R is logically equivalent to ~(P ∧ Q) ∨ R. In other words, if P and Q are both true, then R must be true in order for the statement to be true.

##### Is P → Q ∧ Q → P logically equivalent to P → Q ∨ Q ↔ P?

No, P → Q ∧ Q → P is not logically equivalent to P → Q ∨ Q ↔ P. To see this, consider the truth table for P → Q ∧ Q → P:

P | Q | P → Q | Q → P | P → Q ∧ Q → P
T | T | T | T | T
T | F | F | T | F
F | T | T | T | F
F | F | T | T | T

Now, consider the truth table for P → Q ∨ Q ↔ P:

P | Q | P → Q | Q ↔ P | P → Q ∨ Q ↔ P
T | T | T | T | T
T | F | F | F | F
F | T | T | T | T
F | F | T | T | T

As we can see, the two truth tables are not equivalent.