# Examples of logically equivalent statements

Examples In logic. The following statements are logically equivalent: If Lisa is in Denmark, then she is in Europe (a statement of the form ). If Lisa is not in Europe, then she is not in Denmark (a statement of the form ¬ ¬).

For example,
• P→Q is logically equivalent to ⌝P∨Q. So.
• ⌝(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.
• This means that ⌝(P→Q) is logically equivalent toP∧⌝Q.
Apr 17, 2022

## What does logically equivalent mean in logic?

In basic propositional logic, you represent a statement by a letter, for example p, q, r, etc. Stating, in a formal system, that two statements p and q are logically equivalent is basically saying that the terms used in these statements are different but that p and q mean the same thing.

## What is an equivalent statement of the form?

Therefore, an equivalent statement would be of the form. Hence, we would say, Henry, is a teacher or Paulos is not an accountant. Furthermore, there are times when we would instead state reasons for why two statements are logically equivalent, rather than constructing a truth table.

## How do you determine the equivalence of two logical statements?

There are two possible ways to examine the equivalence of two logical statements, The first method: is considering the value of the logical statement for every possible input; usually in the form of a truth table.

## What is the equivalent of if a then B?

This is an often useful equivalence: “If A, then B” is equivalent to the statement “If not B, then not A”. We saw in the last section that negation of the statement “If A, then B” is the equivalent to the statement “A and not B”.

## How do you know if a statement is logically equivalent?

Two statement forms are logically equivalent if, and only if, their resulting truth tables are identical for each variation of statement variables. p q and q p have the same truth values, so they are logically equivalent.

## Which is logically equivalent to P ∧ Q → R?

(p ∧ q) → r is logically equivalent to p → (q → r).

## What do you mean by logical equivalence explain with an example?

Logical equivalence is a type of relationship between two statements or sentences in propositional logic or Boolean algebra. The relation translates verbally into “if and only if” and is symbolized by a double-lined, double arrow pointing to the left and right ( ).

## Which statement is logically equivalent to Q → P?

A conditional statement is logically equivalent to its contrapositive. Converse: 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.

## Are P → r ∨ 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.

∴∼(∼p⇒q)≡∼p∧∼q.

## Is P ↔ q equivalent to P ↔ q justify?

Namely, p and q are logically equivalent if p ↔ q is a tautology.

## How do you write 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. 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 are equivalent statements?

Equivalent Statements are statements that are written differently, but hold the same logical equivalence. Case 1: “ If p then q ” has three equivalent statements. RULE.

## Which statement is logically equivalent to P → q quizlet?

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

## What is an example of a conditional statement?

Example: We have a conditional statement If it is raining, we will not play. Let, A: It is raining and B: we will not play. Then; If A is true, that is, it is raining and B is false, that is, we played, then the statement A implies B is false.

## 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.

## What is the converse of P → q?

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

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

The proposition p ∧ ( ∼ p ∨ q ) is: a tautology. logically equivalent to. logically equivalent to….Subscribe to GO Classes for GATE CSE 2023.tagstag:appleforce match+appleviewsviews:100scorescore:10answersanswers:26 more rows•Apr 22, 2021

## 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 is the truth table of p λ 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….Truth Tables.pqp→qTFFFTTFFT1 more row

## How many ways can you prove that two statements are logically equivalent?

There are an infinite number of ways to prove that two statements are logically equivalent.

## What is logical equivalence?

In traditional or classical propositional or predicate logic, logical equivalence is synonymous with material equivalence, which basically means the two claims in question have the same truth-value — i.e., they’re both true or both false. In this sense, material or logical equivalence is an assertion about the semantics of the two statements.

## What is a discourse in logic?

A discourse in logic is about trying to establish viable prerequisites and ponder on logical outcomes thereof.

## What is the meaning of “if only if”?

The phrase “if and only if” is usually the way material / logical equivalence is expressed in English. The only assumption justified by an assertion of material or logical equivalence is that the two statements are claimed to have the same truth-value.

## Can logic be used to falsify prerequisites?

Logic can be used to falsify prerequisites too – if you suppose a prerequisite and the logical outcome, and see the outcome is different, this may show that the presumed prerequisites are at the very least incomplete.

## Is a proposition substitutable?

Lastly, as assertions or propositions, they are substitutable for each other.

## Is a statement true if the result follows the given prerequisites?

Statements are logically true if the result follows the given prerequisites under the assumption mentioned prerequisites are accurate. Bear in mind, this makes for an interesting loophole wherein should the prerequisite be false (even outrageously so), the logic may still be sound, even if you can still judge the whole excercise illogical knowing a fuller extent of the subject. Case en point:

## Why are statements logically equivalent?

While the statements may be contradictory, they are logically equivalent because they are covering the same topic. The statements can be derived from one another to determine which is true, which are false, or if both are true or false, based on the topic. The following table depicts how two statements that are logically equivalent correlate …

## How to determine if two sentences are logically equivalent?

A great tool to utilize when determining if two sentences are logically equivalent is analytical reasoning. Analytical reasoning looks at a set of statements, determines the facts and situations associated with those statements and determines what is true or not. Some ways to use analytical reasoning when studying two statements would be;

## What is logical equivalence?

Logical equivalence occurs when two statements have the same truth value. This means that one statement can be true in its own context, and the second statement can also be true in its own context, they just both have to have the same meaning. In the provided example, the partners are arguing about the same fruit and also about it’s color.

## Why do both statements have the same meaning?

Both statements have the same meaning or coverage because both statements have to do with passing the exam. Again, neither statement proves one or the other false, so they are logically equivalent. John thinks Bob is happy he has a car, while Haley thinks Bob is happy because he has a house.

## How to use analytical reasoning when studying two statements?

Some ways to use analytical reasoning when studying two statements would be; Read each statement carefully to determine the truth value, read each statement as its own independent entity, and highlight text or truth diagrams (i.e. truth tables).

## Why does your partner think the fruit is not new and exotic?

You partner, on the other hand, thinks the fruit is not new and exotic because he does not think the color is blue. Which of you has the right perception? The right answer is that neither of you are right or wrong, both of your statements are true in their own right.

## What is an equivalent statement in math?

Take for example the statement “If is even, then is an integer.” An equivalent statement is “If is not an integer, then is not even.” The original statement had the form “If A, then B” and the second one had the form “If not B, then not A.” (Here A is the statement ” is even”, so “not A” is the statement ” is not even”, and B is the statement ” is an integer” so “not B” is the statement ” is not an integer.”)

## What is the equivalent of “If A, then B”?

This is an often useful equivalence: “If A, then B” is equivalent to the statement “If not B, then not A”.

## What does it mean when a set of statements is inconsistent?

Saying that a set of statements is inconsistent means that there is no way they can all be true simultaneously.

## Is a set inconsistent?

So the set is inconsistent.