Converses, inverses, and contrapositives

Learning Objectives Define conditional, converse, inverse, and contrapositive statements. Write the converse, inverse, and contrapositive of a given conditional statement. Translate conditional statements and their related forms into symbolic notation (e.g., p → q). Determine the truth value of a conditional, converse, inverse, and contrapositive statement. Identify the logical equivalence between a conditional statement and its contrapositive. Identify the logical equivalence between the converse and the inverse of a statement. If you get a 90% on the test, then you get an A. 🤔 Does that mean if you get an A, you must have scored 90%? This lesson explores how to rearrange and modify 'if-then' statements to form their converse, inverse, and contrapositive. Unders...

TermDefinitionExample Conditional StatementA logical statement with two parts, a hypothesis and a conclusion, typically written in 'if-then' form. It is the starting point for the other forms.If a polygon has three sides, then it is a triangle. Hypothesis (p)The 'if' part of a conditional statement. It represents the condition or premise.In 'If it is raining, then the ground is wet,' the hypothesis is 'it is raining'. Conclusion (q)The 'then' part of a conditional statement. It represents the result or outcome.In 'If it is raining, then the ground is wet,' the conclusion is 'the ground is wet'. ConverseFormed by switching the hypothesis and the conclusion of the conditional statement.The converse of 'If a figure is...

Symbolic Forms of Logical Statements Conditional: \(p \rightarrow q\) Converse: \(q \rightarrow p\) Inverse: \(\sim p \rightarrow \sim q\) Contrapositive: \(\sim q \rightarrow \sim p\) Use these symbols to represent the logical structure of statements. 'p' is the hypothesis, 'q' is the conclusion, '→' means 'implies' (if-then), and '~' means 'not' (negation). The Law of Logical Equivalence Conditional \(\equiv\) Contrapositive: \((p \rightarrow q) \equiv (\sim q \rightarrow \sim p)\) Converse \(\equiv\) Inverse: \((q \rightarrow p) \equiv (\sim p \rightarrow \sim q)\) A conditional statement is always logically equivalent to its contrapositive (they are either both true or both false). Similarly, the converse is alw...