# When a condition in an IF THEN statement is true?

## When a condition in an IF THEN statement is true?

In the truth table above, p q is only false when the hypothesis (p) is true and the conclusion (q) is false; otherwise it is true. Note that a conditional is a compound statement. Now that we have defined a conditional, we can apply it to Example 1….Definition: A Conditional Statement is…

p | q | p q |
---|---|---|

F | T | T |

F | F | T |

## How do you determine if a statement is true or false?

A statement is true if what it asserts is the case, and it is false if what it asserts is not the case. For instance, the statement “The trains are always late” is only true if what it describes is the case, i.e., if it is actually the case that the trains are always late.

**Why is a conditional statement true when the hypothesis is false?**

A conditional statement that is true by virtue of the fact that its hypothesis is false is often called vacuously true or true by default. Thus the statement “If you show up for work Monday morning, then you will get the job” is vacuously true if you do not show up for work Monday morning.

### When p is false and q is true then p or q is true?

A second style of proof is begins by assuming that “if P, then Q” is false and derives a contradiction from that. In the truth tables above, there is only one case where “if P, then Q” is false: namely, P is true and Q is false….IF…., THEN….

P | Q | If P, then Q |
---|---|---|

F | T | T |

F | F | T |

### What does P and Q mean in logic?

Suppose we have two propositions, p and q. The propositions are equal or logically equivalent if they always have the same truth value. That is, p and q are logically equivalent if p is true whenever q is true, and vice versa, and if p is false whenever q is false, and vice versa.

**What is logically equivalent to P and Q?**

A compound proposition that is always True is called a tautology. Two propositions p and q are logically equivalent if their truth tables are the same. Namely, p and q are logically equivalent if p ↔ q is a tautology. If p and q are logically equivalent, we write p ≡ q.

#### What does P ∧ Q mean?

P ∧ Q means P and Q. P ∨ Q means P or Q. An argument is valid if the following conditional holds: If all the premises are true, the conclusion must be true. So, when you attempt to write a valid argument, you should try to write out what the logical structure of the argument is by symbolizing it.

#### What does P if Q mean?

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.

**What is the truth value of P ∨ Q?**

The disjunction of p and q, denoted by p ∨ q, is the proposition “p or q.” The truth value of p ∨ q is false if both p and q are false. Otherwise, it is true.

## What is P and Q in truth table?

They are used to determine the truth or falsity of propositional statements by listing all possible outcomes of the truth-values for the included propositions. Given two propositions, p and q, “p and q” forms a conjunction. The conjunction “p and q” is only true if both p and q are true.

## Is tautology a P or PA?

So, “if P, then P” is also always true and hence a tautology. Second, consider any sentences, P and Q, each of which is true or false and neither of which is both true and false….P and Not(P)

P | Not(P) | P and Not(P) |
---|---|---|

T | F | F |

F | T | F |

**Which is the inverse of P → Q?**

The inverse of p → q is ∼ p →∼ q. A conditional statement and its converse are NOT logically equivalent.

### What is the converse of p => q?

In logic and mathematics, the converse of a categorical or implicational statement is the result of reversing its two constituent statements. For the implication P → Q, the converse is Q → P. For the categorical proposition All S are P, the converse is All P are S.

### Which is the Contrapositive of P → Q?

Contrapositive: The contrapositive of a conditional statement of the form “If p then q” is “If ~q then ~p”. Symbolically, the contrapositive of p q is ~q ~p. A conditional statement is logically equivalent to its contrapositive.

**Which is the inverse of P → Q quizlet?**

If p = a number is negative and q = the additive inverse is positive, the original statement is p → q. If p = a number is negative and q = the additive inverse is positive, the inverse of the original statement is ~p → ~q.

#### Which Biconditional statement is true?

Definition: A biconditional statement is defined to be true whenever both parts have the same truth value. The biconditional operator is denoted by a double-headed arrow . The biconditional p q represents “p if and only if q,” where p is a hypothesis and q is a conclusion.

#### Where p and q are statements p q is called the of P and Q?

The statement p is called the hypothesis of the implication, and the statement q is called the conclusion of the implication. The biconditional or double implication p ↔ q (read: p if and only if q) is the statement which asserts that p and q if p is true, then q is true, and if q is true then p is true.

**Is the following conditional statement true or false if the grass is red then the snow is white?**

q = the grass is red. with p true and q false. The conditional statement is false only if p is true and q is false. Then, the truth value of “If the snow is white, then the grass is red” is false.

## What is IF AND THEN statement?

A conditional statement (also called an If-Then Statement) is a statement with a hypothesis followed by a conclusion. The hypothesis is the first, or “if,” part of a conditional statement. The conclusion is the second, or “then,” part of a conditional statement.

## What is the truth value for the following conditional statement P false Q true?

The truth value for the following conditional i.e., conjunction statement P is false and Q is true is False.

**What is a Contrapositive in logic?**

: a proposition or theorem formed by contradicting both the subject and predicate or both the hypothesis and conclusion of a given proposition or theorem and interchanging them “if not-B then not-A ” is the contrapositive of “if A then B “

### What is Contrapositive example?

Switching the hypothesis and conclusion of a conditional statement and negating both. For example, the contrapositive of “If it is raining then the grass is wet” is “If the grass is not wet then it is not raining.”

### Can a Contrapositive be false?

Truth. If a statement is true, then its contrapositive is true (and vice versa). If a statement is false, then its contrapositive is false (and vice versa). If a statement (or its contrapositive) and the inverse (or the converse) are both true or both false, then it is known as a logical biconditional.

**How do you prove Contrapositive?**

In mathematics, proof by contrapositive, or proof by contraposition, is a rule of inference used in proofs, where one infers a conditional statement from its contrapositive. In other words, the conclusion “if A, then B” is inferred by constructing a proof of the claim “if not B, then not A” instead.

#### How do you prove Implications?

Direct Proof

- You prove the implication p –> q by assuming p is true and using your background knowledge and the rules of logic to prove q is true.
- The assumption “p is true” is the first link in a logical chain of statements, each implying its successor, that ends in “q is true”.

#### Is Contrapositive the same as Contrapositive?

As nouns the difference between contrapositive and contraposition. is that contrapositive is (logic) the inverse of the converse of a given proposition while contraposition is (logic) the statement of the form “if not q then not p”, given the statement “if p then q”.

**Why does Contrapositive proof work?**

So, in proof by contraposition we assume that is false and then show that is false. It differs from proof by contradiction in the sense that, in proof by contradiction we assume to be false and to true and show that such an assumption leads to something which is known to be false .

## Is Converse and Contrapositive logically equivalent?

Contrapositive proofs work because if the contrapositive is true, due to logical equivalence, the original conditional statement is also true. The contrapositive of this statement is “If not P then not Q.” Since the inverse is the contrapositive of the converse, the converse and inverse are logically equivalent.

## How do you negate a Contrapositive?

To form the inverse of the conditional statement, take the negation of both the hypothesis and the conclusion. The inverse of “If it rains, then they cancel school” is “If it does not rain, then they do not cancel school.”…Converse, Inverse, Contrapositive.

Statement | If p , then q . |
---|---|

Contrapositive | If not q , then not p . |