Biconditionals

Learning Objectives Define a biconditional statement and identify its key phrase, 'if and only if'. Deconstruct a biconditional statement into its two component conditional statements (the conditional and its converse). Write a single biconditional statement by combining a true conditional statement and its true converse. Determine the truth value of a biconditional statement by analyzing the truth values of its component parts. Translate between symbolic notation (p ↔ q) and written biconditional statements. Apply the concept of biconditionals to evaluate and write precise definitions in geometry. Is the statement 'If a polygon has four sides, then it is a quadrilateral' a two-way street? Can we reverse it and have it still be true? Let's find out!...

TermDefinitionExample StatementA declarative sentence that can be objectively determined as either true or false. We often represent statements with letters like p and q.p: 'A triangle has three sides.' (True) Conditional StatementAn 'if-then' statement that connects two statements, a hypothesis (p) and a conclusion (q). It is written as p → q.If an angle measures 90 degrees (p), then it is a right angle (q). ConverseThe statement formed by swapping the hypothesis and conclusion of a conditional statement. The converse of p → q is q → p.The converse of the previous example is: 'If an angle is a right angle (q), then it measures 90 degrees (p).' Biconditional StatementA single statement that combines a conditional and its converse with the phrase 'if and...

Symbolic Form of a Biconditional p ↔ q This is the symbolic representation of a biconditional statement. It is read as 'p if and only if q'. This implies that p is true when q is true, and p is false when q is false. Equivalence to Two Conditionals (p ↔ q) ≡ (p → q) ∧ (q → p) A biconditional statement is logically equivalent to the conjunction (the 'and') of a conditional statement and its converse. For the biconditional to be true, both the conditional (p → q) and its converse (q → p) must be true. Truth Table for Biconditionals A biconditional p ↔ q is TRUE only when p and q have the same truth value (both true or both false). Use this rule to quickly determine the truth value. If p is true and q is false (or vice-versa), the biconditional is fa...