An experiment SETS AND LOGIC SENTENTIAL LOGIC SENTENTIAL

An experiment…. . SETS AND LOGIC

SENTENTIAL LOGIC �

SENTENTIAL LOGIC �

ANOTHER EXAMPLE �

LOGICAL SYMBOLS �

LOGICAL SYMBOLS �

REVISITING �

MORE EXAMPLES: DISCUSSION… WRITE AS LOGICAL EXPRESSIONS � Either John went to the store or we’re out of eggs � Joe is going to leave home and not come back � Either Bill is at work and Jane isn’t, or Jane is at work and Bill isn’t

REVIEW 1: �

WRITE AS SENTENCES �

SECTION 2: TRUTH TABLES �

FROM SENTENCES TO TRUTH TABLES � Premises: � It will either rain or snow tomorrow � It’s too warm for snow � Conclusion � It will rain � Show this with the truth table

SENTENCES TO TRUTH TABLES � Premises � Either John isn’t stupid and he is lazy, or he’s stupid � John is stupid � Conclusion: Therefore John isn’t lazy? ? ?

SENTENCES TO TRUTH TABLES � Premises: � The butler and the cook are not both innocent � Either the butler is lying or the cook is innocent � Conclusion � The butler is either lying or guilty

EQUIVALENT STATEMENTS �

LOGICAL LAWS �

USING THE LOGICAL LAWS �

TAUTOLOGIES AND CONTRADICTIONS �

TAUTOLOGIES AND CONTRADICTIONS �

SIMPLIFYING FORMULAS �

VARIABLES AND SETS � We will consider statements dependent upon a variable or a number of variables � Ex’s � P(x): x is a prime number � D(x, y): x is divisible by y � In this case we don’t have truth tables… we have truth sets

VARIABLES AND SETS � Consider P(x), the statement that x is a prime number � Consider D(x, y), the statement that x is divisible by y � Given x, what is the truth set of D(x, y) if P(x) is true?

EXAMPLES: USE VARIABLES AND SETS � x is a prime number and either y or z is divisible by x � x is a man and y is a woman and x likes y and y doesn’t like x

SET NOTATION �

SET NOTATION AND TRUTH SETS �

SET NOTATION AND TRUTH SETS �

MORE SETS AND TRUTH SETS �

OPERATIONS ON SETS �

OPERATIONS ON SETS AND LOGICAL OPS �

A NEW SET THEORY IDENTITY �

CONDITIONAL STATEMENTS �

EQUIVALENCES �

THE CONVERSE AND CONTRAPOSITIVE �