Understanding the if and only if (biconditional) statement in logic | implications and converses

When the original statement and converse are both true.So, p->q is true and q->p is true. It can be written as p if and only if q”

The phrase “p if and only if q” is a shorthand way of expressing that both the implication (p -> q) and its converse (q -> p) are true

The phrase “p if and only if q” is a shorthand way of expressing that both the implication (p -> q) and its converse (q -> p) are true. This is also known as a biconditional statement.

In order to determine if a statement can be written as “p if and only if q”, we need to check if both the original statement (p -> q) and its converse (q -> p) are true.

In logic, an implication (p -> q) is a statement that asserts that if p is true, then q must also be true. The converse (q -> p) is the statement that asserts that if q is true, then p must also be true.

For a statement to be true if and only if (p if and only if q), both the implication and its converse should be true.

To illustrate this with an example:
Let’s say the original statement is: “If it is raining, then the ground is wet.” (p -> q)
And its converse is: “If the ground is wet, then it is raining.” (q -> p)

If it is indeed the case that whenever it’s raining, the ground is wet, and whenever the ground is wet, it is raining, then both the implication and its converse are true. In this scenario, we can write the statement as “It is raining if and only if the ground is wet” or “Raining <-> Ground is wet” (using the biconditional symbol <->).

However, if the implication (p -> q) is true but the converse (q -> p) is false, or vice versa, then the statement cannot be written as “p if and only if q”.

In summary, “p if and only if q” means that both the implication (p -> q) and its converse (q -> p) are true. This notation is used to indicate that two statements are equivalent and can be used interchangeably.

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