Bell Work 1) Find the value of the variables 2)Write the conditional and converse of each biconditional, and state if the biconditional is true or false. ***Remember what makes a biconditional statement true. a) You go to Herron if and only if you live in Indianapolis b) Two angles are complementary if and only if they add to 90° c) Two segments are congruent if and only if they have the same measure.

Outcomes I will be able to: 1) Identify the inverse, converse, and contrapositive of a conditional statement 2) Use symbolic notation to represent logical statements 3) Form conclusions by applying laws of logic

Logic Puzzle Determine where each person was born and what their favorite hobby is. Fill in the grid as you go. Shade in the boxes that you know cannot be the answer. Leave Blank the ones that must be true. Once you are finished, check with another group to determine if you are correct or not. Then answer the questions at the bottom of the paper.

Conditional Statements Conditional statements may be written in symbols(symbolic notation) where p represents the hypothesis and q represents the conclusion. For example: Conditional Statement: If the sun is out, then the weather is good p q We can rewrite the conditional as: If p, then q Or it can be written just in symbols as: p --> q

Conditional Example Example 1: : If the Colts have a successful season, then they might have a chance at another Super Bowl title. What does p represent? What does q represent? We can write the above statement symbolically as follows __________ or p --> q The symbol → is read ______. If p, then q then

Example 2 Write the converse of the statement in #1. If the Colts have a successful season, then they might have a chance at another Super Bowl title. What would the symbolic notation be for the converse of the conditional: If p, then q? Converse: If the Colts have a chance at another Super Bowl title, then they have had a successful season. Or just in symbols as: If q then p or q --> p q p

Inverse and Contrapositive Symbols In order to write the inverse and contrapositive we need a symbol for negation The symbol for negation is ~, and is written before the letter. ***It is read as not.

Example If we have the statement “if Angle A measures 80°, then Angle A is acute,” we have the following symbols: p q ~ ~ ~p ~q

Example The inverse symbolically would be: If ~p then ~q, or ~p --->~q It would read: If Angle A is not 80°, then Angle A is not Acute. The converse symbolically would be: If q then p, or q --->p. It would read: If Angle A is acute, then Angle A is 80° The contrapositive would be: If ~q then ~p, or ~q ---> ~p It would read: If Angle A is not acute, then Angle A is not 80° Which of these three statements are true? The contrapositive

On Your Own Try the example in your notes on your own. Let p be “You come in early” and q be “Mrs. Stall will help you with Geometry.” Conditional statement: p ---> q a) Write the contrapositive b) Write the inverse I will be coming around to check on your work

Example Using p and q below, write the symbolic statement. p: The angle has a measure of 120°. q: The angle is not acute. p→q ~q ~p q→p ~q→~p ~p→~q

Deductive Reasoning Logical Argument: Deductive reasoning - using facts, definitions, and accepted properties in a logical order to write a logical argument. Note: This differs from inductive reasoning (as used in Chapter 1). Inductive reasoning used patterns to make conjectures. Deductive vs. Inductive – Which is Which? a. Andrea knows that Robin is a sophomore and Todd is a junior. All the other juniors that Andrea knows are older than Robin. So, Todd is older than Andrea. b. Andrea knows that Todd is older than Chan. She also knows that Chan is older than Robin. Therefore, Todd is older than Robin.

Deductive Reasoning There are two laws of deductive reasoning: 1) Law of detachment - If p --> q is a true conditional statement and p is true, then we can conclude that q is true Example: Conditional: If two angles form a linear pair, then they are supplementary. Fact: Angle A and Angle B are a linear pair. What can we conclude? Conclusion: Angle A and Angle B are supplementary

Deductive Reasoning 2) Law of Syllogism - If p --> q and q --> r are true conditional statements, then we can conclude that p --> r is true *Will be given 2 statements and can make a 3rd Example: Conditional 1: If a bird is the fastest bird on land, then it is the largest of all birds. Conditional 2: If a bird is the largest of all birds, then it is an ostrich What can we conclude? Conclusion: If a bird is the fastest bird on land, then it is an ostrich

Examples Examples: 1. Determine if statement (3) follows from (1) and (2) by the Law of Detachment or the Law of Syllogism. If it does, state which law was used. If it does not, write invalid. (a) (1) If an angle A measures more than 90°, then it is not acute. (2) (3) is not acute.

Examples (b) (1) If you wear the school colors, then you have school spirit. (2) If you have school spirit, then the team feels great. (3) If you wear the school colors, then the team will feel great.

Examples (c) (1) If you eat too much candy, then you will get sick. (2) Naomi got sick. (3) Naomi ate too much candy.

Biconditional Statements Symbolically Biconditional: P if and only if q. Remember, if the biconditional is true, then the conditional and the converse are both true In symbols p q ***Arrow goes both ways because the biconditional is both the conditional and the converse

Exit Ticket 1. If the statements p --> q and q --> r are true, then the statement p --> r is true by the Law of_______ ? 2. If the statement p --> q is true and p is true, then q is true by the Law of _______?. 3. State whether the following argument uses inductive or deductive reasoning: “If it is Friday, then Kendra’s family has pizza for dinner. Today is Friday, therefore, Kendra’s family will have pizza for dinner.” 4. Given the notation for a conditional statement is p --> q, what statement is represented by q --> p?