CHAP 1: L OGIC “If it was so, it might be: and if it were so, it would be; but as it isn’t, it ain’t. That’s logic” - Lewis Carroll, author of Alice in Wonderland

P ART 1: S TATEMENTS & S TANDARD F ORMS Statement: Any declarative sentence that is either true (T) or false (F) but not both T or F are called truth value of the statement Statements use lower case letters like p, q, r

EXAMPLES OF STATEMENTS 1. Let p be the sentence “Santa is Jolly” 2. Let q be the sentence “ Christmas is on July 4 th ” Number 1 & 2 are statements bc we can assign to them a truth value Is the following a statement? “Solve the following for X” THE ABOVE is a COMMAND

M ORE W ORDS Negation The statement whose truth value is the opposite of the truth value of p The symbol for a negation is ~ When you have ~ p it is read as “not p”

EXAMPLES OF NEGATION STATEMENTS 1. Let p be the sentence “Santa is Jolly” What would be a ~p ? “Santa is not jolly” 2. Let q be the sentence “ Christmas is on July 4 th ” What would be a ~q ? “Christmas is not on July 4 th ” Notice that these negations are also statements that can be given truth values

T RUTH T ABLES Used to show the truth values for different statements Each Column represents a statement And the rows below show the different truth values possible P~ P TF FT Q~ Q

QUICK NOTE The negations of statements will be referred to as being in STANDARD FORM Example: the standard form of the negation of the statement “Santa is jolly” is the statement “Santa is not jolly” The standard form of the negation of the statement “ Texas is not a state” is ……. “Texas is a state”

YOUR TURN TO TRY Write a negation in standard for each of the statements 1. a: Dallas is the capital of Oklahoma 2. b: An orange is a fruit 3. c: An apple is not Blue

C HECK Y O U R A N S W E R S 1. ~a: Dallas is not the capital of Oklahoma 2. ~b: An orange is not a fruit 3. ~c: An apple is Blue

N OW L ETS G IVE THE E ACH S TATEMENT A T RUTH V ALUE a __________ ~ a _________ c __________ ~ c _________ b __________ ~ b __________ 1.a: Dallas is the capital of Oklahoma 2.b: An orange is a fruit 3.c: An apple is not Blue 1.~a: Dallas is not the capital of Oklahoma 2.~b: An orange is not a fruit 3.~c: An apple is Blue

A NOTHER N EW W ORD Conjunction Is putting two statements together using the word and The symbol for and is ^ Example p ^ q is read as “p and q” This statement is true ONLY if BOTH are true

EXAMPLES OF CONJUNCTION STATEMENTS 1. Let p be the sentence “Santa is Jolly” 2. Let q be the sentence “ Christmas is on July 4 th ” 3. Let r be the sentence “George Washington was the first U.S. President: Write a conjunction of p and q Write a conjunction of p and r Write a conjunction of q and r

C HECK YOUR ANSWERS Santa is jolly and Christmas is on July 4 th Santa is jolly and George Washington was the first U.S. President Christmas is on July 4 th and George Washington was the first U.S. President

W HAT ARE THE T RUTH V ALUES ? Santa is jolly and Christmas is on July 4 th FALSE! FALSE! Santa is jolly and George Washington was the first U.S. President TRUE! TRUE! TRUE! Christmas is on July 4 th and George Washington was the first U.S. President FALSE!!! FALSE!

A G ENERAL T RUTH T ABLE FOR P ^ Q pqp ^ q

O NE L AST W ORD DISJUNCTION Is putting two statements using the word or The symbol for or is v Example p v q is read as “ p or q” The statement is true IF and ONLY IF at LEAST ONE of p and q is TRUE

EXAMPLES OF DISJUNCTION STATEMENTS 1. Let p be the sentence “Santa is Jolly” 2. Let q be the sentence “ Christmas is on July 4 th ” 3. Let r be the sentence “George Washington was the first U.S. President: Write a disjunction of p and q Write a disjunction of p and r Write a disjunction of q and r

C HECK YOUR ANSWERS Santa is jolly or Christmas is on July 4 th Santa is jolly or George Washington was the first U.S. President Christmas is on July 4 th or George Washington was the first U.S. President

W HAT ARE THE T RUTH V ALUES ? Santa is jolly or Christmas is on July 4 th TRUE! Santa is jolly or George Washington was the first U.S. President TRUE! TRUE! TRUE! Christmas is on July 4 th and George Washington was the first U.S. President TRUE! TRUE! TRUE!

WHAT ABOUT………..????? ~ p v q

A G ENERAL T RUTH T ABLE FOR P V Q pqp v q

NOW TIME TO PUT IT ALL T OGETHER Given Statements a: Abraham Lincoln was the 16 th U.S. president b: A lemon is tart c: Valentines Day is in December

Q UESTIONS 1. a ^ b 2. a v c 3. ~a ^ b 4. ~ c v ~ a 5. The truth value for # 1 is ____________ 6. The truth value for # 2 is ____________ 7. The truth value for # 3 is ____________ 8. The truth value for # 4 is ____________