1. Problem: Write the negation, converse, inverse, and contrapositive of the implication and determine the truth value for each. 2. Definitions: - Original implication: $p \to q$ - Negation: $p \wedge \neg q$ - Converse: $q \to p$ - Inverse: $\neg p \to \neg q$ - Contrapositive: $\neg q \to \neg p$ 3. a. "If 10 is a prime number then 10 divides 30." - $p$: 10 is prime - $q$: 10 divides 30 Negation: 10 is prime and 10 does not divide 30. Truth: False (10 divides 30) Converse: If 10 divides 30 then 10 is prime. Truth: False (10 divides 30 but 10 is not prime) Inverse: If 10 is not prime then 10 does not divide 30. Truth: False (10 is not prime but 10 divides 30) Contrapositive: If 10 does not divide 30 then 10 is not prime. Truth: True (contrapositive always has same truth as original) Original implication truth: True (10 is not prime but the implication is true because the antecedent is false) 4. b. "A quadrilateral is a square if and only if its diagonals are equal in length." - $p$: Quadrilateral is a square - $q$: Diagonals are equal This is a biconditional $p \leftrightarrow q$. Negation: Quadrilateral is a square and diagonals are not equal, or quadrilateral is not a square and diagonals are equal. Truth: False (both directions are true for squares) Converse: Same as original because biconditional. Truth: True Inverse: If not square then diagonals not equal. Truth: False (some non-square quadrilaterals have equal diagonals) Contrapositive: If diagonals not equal then not square. Truth: True Original biconditional truth: True 5. c. "If adjacent sides of a quadrilateral are equal then the quadrilateral is a rhombus." - $p$: Adjacent sides equal - $q$: Quadrilateral is rhombus Negation: Adjacent sides equal and quadrilateral is not rhombus. Truth: False (if adjacent sides equal but not rhombus is false) Converse: If quadrilateral is rhombus then adjacent sides equal. Truth: True Inverse: If adjacent sides not equal then quadrilateral is not rhombus. Truth: True Contrapositive: If quadrilateral is not rhombus then adjacent sides not equal. Truth: True Original implication truth: True Final answers: For a: - Negation: False - Converse: False - Inverse: False - Contrapositive: True - Original: True For b: - Negation: False - Converse: True - Inverse: False - Contrapositive: True - Original: True For c: - Negation: False - Converse: True - Inverse: True - Contrapositive: True - Original: True