# In symbolic logic why (p->q) and (p->~q) don’t contradict each other?

## What does P → Q mean in math?

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.

## How do you read Pvq?

(p v q) is a proposition, call it r, so read ~(p v q) as “it is not the case that the proposition r is true”. p and q are also propositions, so e.g. ~p is the proposition “it is not the case that p”. Read [(~p) v (~q)] as “it is the case that either (it is not the case that p) or (it is not the case that q).

## How do you negate an OR statement?

One thing to keep in mind is that if a statement is true, then its negation is false (and if a statement is false, then its negation is true).
Summary.

Statement Negation
“A or B” “not A and not B”
“A and B” “not A or not B”
“if A, then B” “A and not B”
“For all x, A(x)” “There exist x such that not A(x)”

## What is the logical equivalent of P ↔ q?

⌝(P→Q) is logically equivalent to ⌝(⌝P∨Q). Hence, by one of De Morgan’s Laws (Theorem 2.5), ⌝(P→Q) is logically equivalent to ⌝(⌝P)∧⌝Q.

## What is 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.”

## Is P → Q ↔ P a tautology a contingency or a contradiction?

Two compound propositions, P and Q, are said to be logically equivalent if and only if the proposition P ↔ Q is a tautology. The assertion that P is logically equivalent to Q will be expressed symbolically as “P ≡ Q”. For example, (p → q) ≡ (¬p ∨ q), and p ⊕ q ≡ (p ∨ q) ∧ ¬(p ∧ q).

## Which of the following is a contradiction?

(p∧q)∧∼(p∨q) is a contradiction.

## Is PQ equivalent to (~ P → Q justify?

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.

## Are the statements P → Q ∨ R and P → Q ∨ P → are 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.

∴∼(∼p⇒q)≡∼p∧∼q.

## Which of the following propositions is tautology Pvq → Qpv Q → P PV P → Q Both B & C?

The correct answer is option (d.) Both (b) & (c). Explanation: (p v q)→q and p v (p→q) propositions is tautology.

## Which of the proposition is p (~ Pvq is a Tautulogy a contradiction logically equivalent to p q All of above?

Which of the proposition is p^ (~ p v q) is

2) A tautulogy
3) All of above
4) Logically equivalent to p ^ q
5) NULL

## Which of the following is not a contradiction Mcq?

4. If A is any statement, then which of the following is not a contradiction? Explanation: A ∨ F is not always false. Explanation: Definition of contingency.

## 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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## What can we directly say about proposition P1?

What can we correctly say about proposition P1:P1 : (p v Ë¥q) ^ (q →r) v (r v p)

• P1 is satisfiable.
• P1 is tautology.
• If p as true and q is true and r is false, then P1 is true.
• If p is true and q is false and r is false, the P1 is true.

## Which of the following pairs of propositions are not logically equivalent?

Hence the above statement is True, Logically not equivalent. ∴ Hence the correct answer is ((p ∧ q) → r ) and ((p → r) ∧ (q → r)).

## How do you know if two propositions are logically equivalent?

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. If p and q are logically equivalent, we write p = q.

## Which pair is logically equivalent?

Two logical statements are logically equivalent if they always produce the same truth value. Consequently, p≡q is same as saying p⇔q is a tautology. Beside distributive and De Morgan’s laws, remember these two equivalences as well; they are very helpful when dealing with implications. p⇒q≡¯q⇒¯pandp⇒q≡¯p∨q.

## What is not logically equivalent?

A conditional statement is not logically equivalent to its inverse. Only if : p only if q means “if not q then not p, ” or equivalently, “if p then q.” Biconditional (iff):

## How do you prove logical equivalence?

Two logical statements are logically equivalent if they always produce the same truth value. Consequently, p≡q is same as saying p⇔q is a tautology.

## What is the negation of P → Q?

The negation of “P and Q” is “not-P or not-Q”.

## Can two false sentences be logically equivalent?

No two false sentences are logically equivalent. circumstances. A pair of equivalent sentences must both be false at the same time if they are false at all.

## What is a logical contradiction?

A logical contradiction is the conjunction of a statement S and its denial not-S. In logic, it is a fundamental law- the law of non contradiction- that a statement and its denial cannot both be true at the same time. Here are some simple examples of contradictions. 1. I love you and I don’t love you.

## Are all logically indeterminate sentences logically equivalent?

A sentence is logically indeterminate just if it is neither logically true nor logically false. An argument is valid just if it is not possible for the premises to be true and the conclusion false. This may be the case when the conclusion is logically indeterminate.

## When two sentences are never both true at the same time those two sentences are said to be?

If two sentences cannot be both true at the same time (form an inconsistent set), they are said to be contrary.

## What is contradiction in truth table?

Contradiction A statement is called a contradiction if the final column in its truth table contains only 0’s. Contingency A statement is called a contingency or contingent if the final column in its truth table contains both 0’s and 1’s.

## Is contradiction always false?

A contradiction is something that is always false, regardless of it’s truth values.