# Are p → q and p ∨ q logically equivalent

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## What 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 P → Q → R and P → Q → R logically equivalent?

No, compare (p⟹p)⟹r, which is equivalent to r, and p⟹(p⟹r), which is equivalent to p⟹r. It is not. Suppose p, q, r are false. Then p→q and q→r are true, so

**(p→q)→r is false**and p→(q→r) is true.## Are the statements P → Q ∨ R and P → Q ∨ P → are logically equivalent?

1.3. 24 Show that

**(p → q) ∨ (p → r)**and p → (q ∨ r) are logically equivalent. By the definition of conditional statements on page 6, using the Com- mutativity Law, the hypothesis is equivalent to (q ∨ ¬p) ∨ (¬p ∨ r). … This means that the conditional from the second-to-last column the last column is always true (T).## How do you know if two statements are logically equivalent?

To test for logical equivalence of 2 statements,

**construct a truth table that includes every variable to be evaluated**, and then check to see if the resulting truth values of the 2 statements are equivalent.## Is P → Q ∨ q → p a tautology?

Example: The

**proposition p ∨ ¬p is a tautology**. 2. A proposition is said to be a contradiction if its truth value is F for any assignment of truth values to its components. … A proposition of the form “if p then q” or “p implies q”, represented “p → q” is called a conditional proposition.## Is P → Q ∧ Q → P logically equivalent to P → Q ∨ Q ↔ P?

Look at the following two compound propositions: p → q and q ∨ ¬p. (p → q) and (q ∨ ¬p) are logically equivalent. So (p → q) ↔ (q ∨ ¬p) is a tautology. Thus:

**(p → q)≡ (q ∨ ¬p)**.## What is the truth value of ∼ P ∨ Q ∧ P?

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.

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Truth Tables.

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Truth Tables.

p | q | p∧q |
---|---|---|

T | F | F |

F | T | F |

F | F | F |

## Is logically equivalent to?

Two expressions are logically equivalent provided that

**they have the same truth value for all possible combinations of truth values**for all variables appearing in the two expressions. In this case, we write X≡Y and say that X and Y are logically equivalent.## Is P → Q → P → Q → QA tautology Why or why not?

[p∧(p→q)]→q ≡ F

**is not true**. Therefore [p∧(p→q)]→q is tautology.## Are the statements P ∧ Q ∨ R and P ∧ Q ∨ R logically equivalent?

This particular equivalence is known as De Morgan’s Law. 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.## Is P -> Q equivalent to Q -> p?

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. … Suppose a conditional statement of the form “If p then q” is given. The converse is “If q then p.” Symbolically, the converse of p q is q p.## What is logical equivalence in math?

From Wikipedia, the free encyclopedia. In logic and mathematics, statements and are said to be

**logically equivalent if they have the same truth value in every model.**## What is logical equivalence examples?

Now, consider the following statement: If Ryan gets a pay raise, then he will take Allison to dinner. This means we can also say that

**If Ryan does not take Allison to dinner, then he did not get a pay raise**is logically equivalent.## Which of the following is logically equivalent to if/p then not q?

If p, then not q” is equivalent to “No p are q.” Example: “If something is a poodle, then it is a dog” is a round-about way of saying “All poodles are dogs.

## What is the negation of the statement P → Q ∨ R?

q ∨ r → pD.

**P**∧∼ q ∧∼ r.## Which is the inverse of P → Q?

The inverse of p → q is

**¬p → ¬q**. If p and q are propositions, the biconditional “p if and only if q,” denoted by p ↔ q, is true if both p and q have the same truth values and is false if p and q have opposite truth values. The words if and only if are sometimes abbreviated iff.## Which statement is denoted by if not p then not q?

Comparisons

name | form | description |
---|---|---|

implication | if P then Q | first statement implies truth of second |

inverse | if not P then not Q | negation of both statements |

converse | if Q then P | reversal of both statements |

contrapositive | if not Q then not P | reversal and negation of both statements |

## What does P only if Q mean?

Only if introduces a necessary condition: P only if Q means that the

**truth of Q is necessary**, or required, in order for P to be true. That is, P only if Q rules out just one possibility: that P is true and Q is false.## Is contrapositive the same as Contraposition?

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”.## When the statements p and q both are true then the value of the statement P → Q ∧ Q is?

This has some significance in logic because if two propositions have the same truth table they are in a logical sense equal to each other – and we say that they are logically equivalent. So: ¬p∨(p∧q)≡p→q, or “Not p or (p and q) is equivalent to if p then q.”

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Logically Equivalent Statements.

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Logically Equivalent Statements.

p | q | p→q |
---|---|---|

F | F | T |

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Mar 7, 2021

## What are logically equivalent statements?

Logical Equivalence. Definition. Two statement forms are called logically equivalent

**if**, and only if, they have identical truth values for each possible substitution for their. statement variables.Ads by Google