Conditional Statements

Conditional Statements If-then statements are called conditional statements. The portion of the sentence following if is called the hypothesis. The part following then is called the conclusion. p q (If p, then q)

If it is an apple, then it is a fruit. Hypothesis – It is an apple. Conclusion – It is a fruit. p q

Converse q p The converse statement is formed by switching the hypothesis and conclusion. If it is an apple, then it is a fruit. Converse: If it is a fruit, then it is an apple. The converse may be true or false.

negation – the denial of a statement Ex. “An angle is obtuse.” Negation – “An angle is not obtuse.”

Inverse ~p ~q An inverse statement can be formed by negating both the hypothesis and conclusion. If it is an apple, then it is a fruit. Inverse: If it is not an apple, then it is not a fruit. The inverse may be true or false.

Contrapositive ~q ~p A contrapositive is formed by negating the hypothesis and conclusion of the converse. If it is an apple, then it is a fruit. Contrapositive: If it is not a fruit, then it is not an apple. The contrapositive of a true conditional is true and of a false conditional is false.

If you double the radius of a circle, then you will know the length of the diameter. Write the given statements in symbolic form. If you don’t know the length of the diameter, then you haven’t doubled the radius. If you don’t double the radius of a circle, then you will not know the length of the diameter. If you know the length of the diameter, then you have doubled the radius. Select the statement(s) above that are not valid. Identify the converse, inverse, and contrapositive. Contrapositive Inverse Invalid Converse

Biconditional p  q or pq A biconditional statement is defined to be true whenever both parts have the same truth value. The biconditional operator is denoted by a double-headed arrow  . The biconditional p q represents "p if and only if q," where p is a hypothesis and q is a conclusion. The following is a truth table for biconditional p q. Biconditional p  q or pq A biconditional statement is defined to be true whenever both parts have the same truth value. The biconditional operator is denoted by a double-headed arrow . The biconditional pq represents "p if and only if q," where p is a hypothesis and q is a conclusion..

Biconditional p  q p: a polygon is a triangle q: a polygon has exactly three sides. A polygon is a triangle if and only if it has exactly three sides.

Biconditional p  q When a statement and the converse are both true, the statements can form a biconditional statement. p : 2x + 7 = 19 q : x = 6 2x + 7 = 19  x = 7 2x + 7 = 19 iff x = 7

The following is a truth table for biconditional pq or pq

Law of Detachment Law of Detachment ( also known as Modus Ponens (MP) ) says that if pq is true and p is true, then q must be true. pq The team who wins tonight’s game wins the district championship. CHS won the game the game. Therefore: CHS won the district championship

Law of Syllogism The Law of Syllogism ( also called the Law of Transitivity ) states: if p  q and q r are both true, then p  r is true. p  q : If you like raisins then you like fruit. q r : If you like fruit, then you like sugar. p  r : If you like raisins, then you like sugar.

Law of Contrapositive Why does a chicken cross the road????? A chicken crosses the road, because he wants to get to the other side. Now let’s use the law of contrapositive to make a valid argument. Why does that stupid chicken just sit there??? If a chicken doesn’t want to get to the other side, then he doesn’t cross the road!

Counterexamples If you drive to school, then you have money. If you have money, then you can buy whatever you want. ∴ If you drive to school, then you can buy whatever you want. (Valid/False) ～ I drive to school but I can’t buy my student’s interest in math. I drive to school, but I can’t buy Trump Towers.