## Can the conditional converse inverse and Contrapositive all be true?

The converse of “If it rains, then they cancel school” is “If they cancel school, then it rains.” To form the inverse of the conditional statement, take the negation of both the hypothesis and the conclusion….Converse, Inverse, Contrapositive.

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

Inverse | If not p , then not q . |

Contrapositive | If not q , then not p . |

### Is a conditional and its converse are always true?

If a conditional and it’s converse are always true, the statement is called a biconditional.

#### What is true about a conditional statement?

A conditional asserts that if its antecedent is true, its consequent is also true; any conditional with a true antecedent and a false consequent must be false. For any other combination of true and false antecedents and consequents, the conditional statement is true.

**Are conditional and Contrapositive always true?**

The contrapositive does always have the same truth value as the conditional. If the conditional is true then the contrapositive is true.

**What is inverse conditional statement?**

The inverse of a conditional statement is when both the hypothesis and conclusion are negated; the “If” part or p is negated and the “then” part or q is negated. In Geometry the conditional statement is referred to as p → q. The Inverse is referred to as ~p → ~q where ~ stands for NOT or negating the statement.

## What makes a conditional true?

A conditional is considered true when the antecedent and consequent are both true or if the antecedent is false. When the antecedent is false, the truth value of the consequent does not matter; the conditional will always be true.

### What is the converse inverse and contrapositive of a conditional statement?

The converse of the conditional statement is “If Q then P.” The contrapositive of the conditional statement is “If not Q then not P.” The inverse of the conditional statement is “If not P then not Q.”

#### What is the converse of the conditional statement?

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.

**What is conditional converse inverse and contrapositive?**

We start with the conditional statement “If P then Q.” The converse of the conditional statement is “If Q then P.” The contrapositive of the conditional statement is “If not Q then not P.” The inverse of the conditional statement is “If not P then not Q.”

**What is converse and contrapositive?**

A contrapositive statement changes “if not p then not q” to “if not q to then, not p.” The converse of the conditional statement is “If Q then P.” The contrapositive of the conditional statement is “If not Q then not P.” The inverse of the conditional statement is “If not P then not Q.”

## When is the converse and the contrapositive logically true?

If the statement is true, then the contrapositive is also logically true. If the converse is true, then the inverse is also logically true. Example 1: In the above example, since the hypothesis and conclusion are equivalent, all four statements are true.

### Which is the inverse of the conditional statement if p then Q?

We start with the conditional statement “If P then Q .” The converse of the conditional statement is “If Q then P .” The contrapositive of the conditional statement is “If not Q then not P .” The inverse of the conditional statement is “If not P then not Q .” We will see how these statements work with an example.

#### How does a contrapositive proof of a conditional statement work?

Rather than prove the truth of a conditional statement directly, we can instead use the indirect proof strategy of proving the truth of that statement’s contrapositive. Contrapositive proofs work because if the contrapositive is true, due to logical equivalence, the original conditional statement is also true.

**Which is the converse of the conditional statement?**

The converse of the conditional statement is “If the sidewalk is wet, then it rained last night.”. The contrapositive of the conditional statement is “If the sidewalk is not wet, then it did not rain last night.”.