Table of Contents

- 1 How do you prove p then q?
- 2 Which method is used to prove statements of the form if/p then q or p implies q?
- 3 When to proof P → Q true we proof P true and Q is also true then what type of proof is this *?
- 4 When to proof p q true we proof P false that type of proof is known as?
- 5 Is P → Q → [( P → Q → Q a tautology?
- 6 Is P and not PA tautology?
- 7 Which is true if p and not ( P )?
- 8 What is the meaning of the proposition p then Q?

## How do you prove p then q?

To prove a statement of the form P ⇒ Q by contradiction, assume the assumption, P, is true, but the conclusion, Q, is false, and derive from this assumption a contradiction, i.e., a statement such as “0 = 1” or “0 ≥ 1” that is patently false: Assume P is true, and that Q is false. …

## Which method is used to prove statements of the form if/p then q or p implies q?

Argument | A series of statements . |
---|---|

Symbol for “therefore”, normally used to identify the conclusion of an argument. | |

Modus Ponens | Latin for “method of affirming.” A rule of inference used to draw logical conclusions, which states that if p is true, and if p implies q (p q), then q is true. |

**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.

**What is p and q implies not p?**

Thus, “p implies q” is equivalent to “q or not p”, which is typically written as “not p or q”. This is one of those things you might have to think about a bit for it to make sense, but even with that, the truth table shows that the two statements are equivalent.

### When to proof P → Q true we proof P true and Q is also true then what type of proof is this *?

If the statement q in the implication p –> q is true regardless of the truth value of p, we have a trivial proof.

### When to proof p q true we proof P false that type of proof is known as?

Determine for which positive integers n the statement P(n) must be true if: P(1) is true; for all positive integers n, if P(n) is true then P(n+2) is true….

Q. | A proof that p → q is true based on the fact that q is true, such proofs are known as |
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B. | contrapositive proofs |

C. | trivial proof |

D. | proof by cases |

**When to proof P → Q true we proof P false that type of proof is known as?**

Trivial Proof: If we know q is true then p → q is true regardless of the truth value of p. Vacuous Proof: If p is a conjunction of other hypotheses and we know one or more of these hypotheses is false, then p is false and so p → q is vacuously true regardless of the truth value of q.

**When to proof p implies q we proof P false that type of proof is known as?**

Vacuous Proof If the statement p in the implication p –> q is false then the implication is always true.

## Is P → Q → [( P → Q → Q a tautology?

(p → q) and (q ∨ ¬p) are logically equivalent. So (p → q) ↔ (q ∨ ¬p) is a tautology. We have a number of rules for logical equivalence.

## Is P and not PA tautology?

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. Consider the sentence, “(P and Not(P)) or Q”….P and Not(P)

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

T | F | F |

F | T | F |

**What is not p and q equal to?**

if p is a statement variable, the negation of p is “not p”, denoted by ~p. If p is true, then ~p is false. Conjunction: if p and q are statement variables, the conjunction of p and q is “p and q”, denoted p q….

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

Negations of t and c | ~t c | ~c t |

**When P is true and Q is false?**

Conditional: The conditional of q by p is “If p then q” or “p implies q” and is denoted by p q. It is false when p is true and q is false; otherwise it is true.

### Which is true if p and not ( P )?

The sentence “if [P and Not(P)], then Q” is always true, regardless of the truth values of P and Q. This is the principle that, from a contradiction, anything (and everything) follows as a logical conclusion. The table below explores the four possible cases, but the truth is simpler than that.

### What is the meaning of the proposition p then Q?

p then q” or “p implies q”, represented “p → q” is called a conditional proposition. For instance: “if John is from Chicago then John is from Illinois”. The proposition p is called hypothesis or antecedent, and the proposition q is the conclusion or consequent. Note that p → q is true always except when p is true and q is false.

**Is the word Q always the same as P?**

Consider the sentence, “(P and Not(P)) or Q”. This means exactly the same as Q, because “P and Not(P))” is always false. This principle is more formally explained by the truth table below: note that columns 2 and 5 have the same truth values.

**How to negate IF-THEN statement p implies Q?**

Negating the conditional if-then statement p implies q 1 Using this to negate the statement. This shows that the negation of “p implies q” is “p and not q”. 2 Verifying with a truth table. Although the work above is enough, you can always double check your results using a truth table. 3 Summary. 4 Continue your study of discrete math