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Warm Up: Follow the if If Fred gets up before 7, then his mother will make him breakfast. If his mother makes him breakfast, Fred does not stop at Bojangles. If Fred does not stop at Bojangles then Sue has to text him about their lunch plans. Ms. Newman has Fred’s cell phone because he was reading a text message in class. What do you know? If Fred gets up at 6, what do you know? Reasoning & Proof Chapter 2 2.1 – Conditional Statements 2.2 – Biconditionals Please have your week packet out to fill in notes. Conditional “if-then” statement. Symbolic Form: p q Read as: If p, then q. The part after “if” is the hypothesis. The part after “then” is the conclusion. Ex.1: Identifying the Hypothesis & Conclusion Put a single underline under the hypothesis and a double under the conclusion. a. If you go to RHS, then you live in Durham. b. If today is Wednesday, then tomorrow is Thursday. c. If y – 3 = 5, then y = 8. Ex.2: Writing a Conditional Write each sentence as a conditional. a. A rectangle has 4 right angles. b. A tiger is an animal. c. A square has 4 congruent sides. d. An acute angle measures less than 90 degrees. A conditional can have a truth value of true or false. To show a conditional true, you must show that every time the hypothesis is true, the conclusion is true as well—meaning you must only find a counterexample for which the hypothesis is true, but the conclusion is false. Ex.3: Finding a Counterexample a. If it is February, then there are only 28 days in the month. b. If you live in Jacksonville, then you live in N.C. Ex.4: Using a Venn Diagram Draw a Venn Diagram to illustrate this conditional. If you live in Chicago, then you live in Illinois. Note: The set that makes the hypothesis true, lies inside the set that makes the conclusion true. Converse Switch the hypothesis and the conclusion. Symbolic Form: q p Read as: If q, then p. Ex.5: Writing the Converse Write the converse of the following conditionals. a. If you go to RHS, then you live in Durham. b. If two lines intersect to form right angles then they are perpendicular. Inverse Put nots in front of hypothesis and conclusion. Symbolic Form: ~p ~q Read as: If not p, then not q. Contrapositive Switch the hypothesis and the conclusion and put nots in front of both Symbolic Form: ~q ~p Read as: If not q, then not p. Ex.7: Write the inverse and the contrapositive of the conditional. If you go to RHS, then you live in Durham. Inverse: Contrapositive: Biconditional When a conditional and its converse are true. “if and only if” Symbolic form: Read as: p q and q p as p q p if and only if q Ex.6: Writing a Biconditional Consider the following conditional statements. Write the converse. If the converse is true, combine the statement to form a biconditional. a. If two lines intersect to form right angles, then they are perpendicular. Converse: Biconditional? b. If you are in Ms. Newman’s 3rd or 4th period class this semester, then you are taking H. Geometry. Converse: Biconditional? Inverse Make both the hypothesis and the conclusion negative. If it snows, then we will be out of school. (p q) If it does not snow then we will not be out of school. (~p ~q) Contra-positive Backwards and negative. The converse of the inverse. p q becomes ~q ~p My Note Page List the logic vocabulary and the symbols. Cooldown 1. What is a conditional? 2. What is the symbolic form for contrapositive? 3. What does stand for? 4. Which term goes with q p ?