Interpreting Truth tables (biconditional statements) Lesson 20 Interpreting Truth tables (biconditional statements)

Review Conditional Statements (Less. 17) Original If p, then q. Converse If q, then p. Inverse If ~p, then ~q. Contrapositive If ~q, then ~p. What is special about the original conditional statement and its contrapositive? They are logically equivalent statements We will discover this is also true for the converse and inverse statements

Biconditional Statement A biconditional statement is a combination of the original conditional statement and its converse using “… if and only if …” Original conditional statement If p, then q. Biconditional Statement p if and only if q. A biconditional statement is true only when both the original and converse are true If an animal is a bird, then it has two legs. Write the converse & find its truth value If an animal has two legs, then it is a bird. False, kangaroo Write as a biconditional & find its truth value. Why? An animal is a bird if and only if it has two legs. False, converse is false

Truth Table A truth table is a table that lists all possible combinations of truth values for a hypothesis, conclusion, and the conditional statement(s) they form Copy for homework and tests You will be asked to add columns Take note of the only combination that is false Why is a conditional statement true even when both the hypothesis & conclusion is false? You can conclude anything from a false conclusion (same for the previous line) Hypothesis Conclusion If p, then q. T F

If a polygon is a quadrilateral, then the figure has four sides. Use a truth table to represent the statement. Add to the truth table to show the converse and biconditional. p q If p, then q. If q, then p. p if and only if q.

If a polygon is a quadrilateral, then the figure has four sides. Use a truth table to represent the statement. We know this is a true statement, but row #1 in the truth table shows why. Add to the truth table to show the converse and biconditional. p q If p, then q. If q, then p. p if and only if q. T F

If a polygon is a quadrilateral, then the figure has four sides. Use a truth table to represent the statement. We know this is a true statement, but row #1 in the truth table shows why. Add to the truth table to show the converse and biconditional. p q If p, then q. If q, then p. p if and only if q. T F

If a polygon is a quadrilateral, then the figure has four sides. Use a truth table to represent the statement. We know this is a true statement, but row #1 in the truth table shows why. Add to the truth table to show the converse and biconditional. If a polygon has four sides, then the figure is a quadrilateral. A polygon is a quadrilateral if and only if it has four sides. p q If p, then q. If q, then p. p if and only if q. T F

Compound statement combines 2 statements using and/or. Conjunction Disjunction Uses “and” To be true both p and q must be true p – salt has sodium q – salt has chloride Salt has sodium and chloride. True Uses “or” To be true at least one of p and q must be true p – the light is on q – the room is dark The light is on or the room is dark. True

Conclusion/Questions? Is a true biconditional statement a conjunction or disjunction? Why? Conjunction because a biconditional requires a true p and a true q. Why do you think disjunction is true in more cases than a conjunction? A disjunction only requires that one of the two statements is true, while a conjunction requires that both be true. p q p and q p or q T F