Conditional

Conditional Statements Vocabulary Conditional: A compound sentence formed by using the words if….then. Given the simple sentences p and q, the conditional would be read as if p then q and is written in symbols as. Hypothesis: Also called the antecedent, it is the first part of the conditional represented above by the letter p and usually follows the word if. Conclusion: Also called the consequent, it is the latter part of the conditional represented above by the letter q and usually follows the word then.

Example: Let p represent “You will get an A in Geometry.” (true) Let q represent “I will buy you a new graphing calculator.” (true) “If you get an A in Geometry, then I will buy you a new graphing calculator.”( ) Note: In order for a conditional statement to be false, the hypothesis (p) must lead to a false conclusion (q).

Decide if the conditional statement is true or false. 1)“If you get an A in Geometry then I buy you a new calculator.” 2)“If you get an A in Geometry then I do not buy you a new graphing calculator.” 3)“If you do not get an A in Geometry then I buy you a new graphing calculator.” 4) “If you do not get an A in Geometry then I do not buy you a new graphing calculator.” (true) (false) ( true )

Try These Example 2: Let m represent ”Monday is the first day of the week.” (True) Let w represent “There are 52 weeks in a year.” (True) Let h represent “ An hour has 75 minutes.” (False) For each of the conditionals, write the sentence which it represents and determine the truth value. m → w w → h ~w → m (m w) → h

Hidden Conditionals Often the words “if…then” may not appear in a statement that does suggest a conditional. Instead, the expressions “when” or “in order that” may suggest that the statement is a conditional. Example: Turn the hidden conditional statement into “if…then” statements 1)“When I finish my homework I will go to the movies.” If I finish my homework, then I will go to the movies. 2) “In order to succeed you must work hard.” If you want to succeed, then you must work hard.

Truth Values A conditional statement is true for all cases with the exception of a statement with a true hypothesis and a false conclusion. Note the truth table below. pq TTT TFF FTT FFT

Complete the following truth table pq~p~p pq

What is the Converse, Inverse and Contrapositive of a conditional statement?

Write the converse of the statement in the cartoon: If they send the get-well card, then you will send the proof of purchase The converse of a conditional is formed by reversing its hypothesis and conclusion

You break it, you buy it If you do not break it, then you do not buy it If you break it, then you buy it p: you break itq: you buy it p  q ~p  ~q The inverse of a conditional negates the hypothesis and the conclusion Conditional:

Cats Black cats If I am a black cat, then l am cat If I am not a black cat, then l am not a cat b: I am a black cat c: I am a cat b  c ~b  ~c If I am not a cat, then l am not a black cat ~c  ~b The Contrapositive of a conditional Negates and Reverses the hypothesis and the conclusion

Conditional p  q Converse q  p Inverse ~p  ~q Contrapositive ~q  ~p

Fill in the truth value, make a conclusion about conditionals and contrapositives pq~p~qp  q~q  ~p What do you notice about the conditional and the contrapositive?

A conditional and its contrapositive are Logically equivalent (They always have the same truth value) p  q is the same as ~q  ~p

1)Write a conditional sentence that represents the Venn diagram. 2) Write in symbolic and sentence form a) The converse b) The inverse c) The contrapositive p: The shape is a square q: The shape is a rectangle squares rectangles If the shape is a square, then it is a rectangle