# The difference between “P if Q” and “P only if Q”?

So it’s obviously correct to read P → Q as P only if Q. If, on the other hand, introduces a sufficient condition: P if Q means that the truth of Q is sufficient, or enough, for P to be true as well. That is, P if Q rules out just one possibility: that Q is true and P is false. But that is exactly what Q → P rules out.

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## What is the difference between if and only if and only if?

IF AND ONLY IF, is a biconditional statement, meaning that either both statements are true or both are false. So it is essentially and “IF” statement that works both ways. Note that IF AND ONLY IF is different than simply ONLY IF.

## Is P if and only if q the same as Q if and only if P?

It says that P and Q have the same truth values; when “P if and only if Q” is true, it is often said that P and Q are logically equivalent.
IF AND ONLY IF.

P Q P if and only if Q
T T T
T F F
F T F
F F T

## What do you call a statement P if and only if q?

In conditional statements, “If p then q” is denoted symbolically by “p q”; p is called the hypothesis and q is called the conclusion. For instance, consider the two following statements: If Sally passes the exam, then she will get the job. If 144 is divisible by 12, 144 is divisible by 3.

## What is if and only if statement?

In logic and related fields such as mathematics and philosophy, “if and only if” (shortened as “iff”) is a biconditional logical connective between statements, where either both statements are true or both are false.

## What is the negation of P if and only if q?

The negation of ‘p if and only if q’ is ‘p and not-q, or q and not-p,’ which, as it happens, is semantically equivalent to the exclusive disjunction, ‘p | q. ‘

## What does ↔ mean in math?

Symbol ↔ or ⟺ denote usually the equivalence, commonly known also as “NXOR”, “if and only if” or “iff” for short (see also its Wikipedia page). More precisely p↔q is equal to (p→q)∧(q→p)

## What is the difference between conditional and biconditional?

A conditional statement is of the form “if p, then q,” and this is written as p → q. A biconditional statement is of the form “p if and only if q,” and this is written as p ↔ q.

## What is inverse conditional statement?

The inverse of a conditional statement is when both the hypothesis and conclusion are negated; the “If” part or p is negated and the “then” part or q is negated. In Geometry the conditional statement is referred to as p → q. The Inverse is referred to as ~p → ~q where ~ stands for NOT or negating the statement.

## What’s a biconditional statement?

Description. A biconditional statement is one of the form “if and only if”, sometimes written as “iff”. The statement “p if and only if q” means “p implies q” AND “q implies p”. That is, it is a conjunction of two individual conditional statements.

## Is the converse of an if and only if statement true?

The sentence “If q, then p” is called its converse. The sentence “p if and only if q” means: If p then q and if q then p. In other words, it means that a sentence and its converse are both true.

## What is the truth value of the compound proposition P → Q ↔ P if P is false and Q is true?

Operation Notation Summary of truth values
Negation ¬p The opposite truth value of p
Conjunction p∧q True only when both p and q are true
Disjunction p∨q False only when both p and q are false
Conditional p→q False only when p is true and q is false

## What are the truth values for ~( p ∨ Q?

So because we don’t have statements on either side of the “and” symbol that are both true, the statment ~p∧q is false. So ~p∧q=F. Now that we know the truth value of everything in the parintheses (~p∧q), we can join this statement with ∨p to give us the final statement (~p∧q)∨p.
Truth Tables.

p q p∨q
T F T
F T T
F F F

## Is ~( p q the same as P Q?

~(P&Q) is not the same as (~P&~Q). You can do this for any logic, and it saves a lot of time waiting for answers from StackExchange!

## What does P → Q mean?

The implication p → q (read: p implies q, or if p then q) is the state- ment which asserts that if p is true, then q is also true. We agree that p → q is true when p is false. The statement p is called the hypothesis of the implication, and the statement q is called the conclusion of the implication.

## Is ~( Pvq and PV Q the same?

Well, what does it mean to say not both? It means that either p is false or q is false or they are both false–anyway, p and q can’t both be true at the same time. So ~(p · q) º ~p v ~q. On the other hand, ~(p v q) means it’s not the case that either p or q.

## Which is logically equivalent to P ↔ Q?

P → Q is logically equivalent to ¬ P ∨ Q . Example: “If a number is a multiple of 4, then it is even” is equivalent to, “a number is not a multiple of 4 or (else) it is even.”

## Are the statements P ∧ Q ∨ R and P ∧ Q ∨ R logically equivalent?

Since columns corresponding to p∨(q∧r) and (p∨q)∧(p∨r) match, the propositions are logically equivalent. This particular equivalence is known as the Distributive Law.

## Does P follow from Pvq?

p v q stands for p or q That is: p v q iff at least one of p or q is true. Note that they may both be true. p ↔ q or p ≡ q stands for p iff q That is: p ↔ q iff either both p and q are true or both p and q are false, i.e. p has the same ‘truth value’ as q.

## Is p ∧ p ∨ q )) → QA tautology?

All true ∴ Tautology proved.

## Is P → Q ∨ q → p a tautology?

(p → q) ∧ (q → p). (This is often written as p ↔ q). Definitions: A compound proposition that is always True is called a tautology.

## Which of the proposition is p ∧ P ∨ Q is?

The proposition p∧(∼p∨q) is: a tautology. logically equivalent to p∧q.
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