We believe in providing comprehensive and reliable educational resources to help students improve their academic pursuits.

## Which of the following is not true?

### Solution

Now that we have an understanding of statements and truth tables let’s review the given statements and evaluate their truth values using a truth table:

p | q | p <—> q |

T | T | T |

T | F | F |

F | T | F |

F | F | T |

Analyzing the truth table, we find the following:

### Statement A

The last column of its truth table contains T and F, indicating that it is not a contradiction. Hence, Statement A is not valid.

### Statement B

Applying the concept of negation, we find that the negation of a negation of a statement is indeed the statement itself. Thus, Statement B is true.

### Statement C

The truth table demonstrates that the statement p ? q is not always true, as it produces both T and F values. Therefore, Statement C is not valid.

### Statement D

The last column of its truth table contains T and F, implying that it is not a tautology. Thus, Statement D is not valid.

**FAQs:**

## What is mathematical reasoning?

Mathematical logic uses logical thinking and deducible reasoning to dissect and break delicate problems. It involves understanding and applying mathematical concepts, principles, and relationships to draw logical conclusions.

## What are statements in mathematical reasoning?

In exemplary logic, statements are declarative rulings that can be classified as true or false. They form the basis of logical reasoning and express mathematical ideas, propositions, or claims.

## What is a truth table?

A truth table systematically represents compound statements’ truth values by evaluating their component statements’ truth values under different combinations. It helps analyze logical relationships, identify patterns, and make deductions based on the truth values.

## What are the tautologies and contradictions in mathematical reasoning?

A tautology is a compound statement that is always true, regardless of the truth values of its component statements. On the other hand, a contradiction is an emulsion statement that’s always false. Tautologies and contradictions play an essential role in mathematical reasoning and logical proofs.