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...
TRANSCRIPT
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CHAP 1: LOGIC
“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
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PART 1: STATEMENTS & STANDARD FORMS
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
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EXAMPLES OF STATEMENTS
1. Let p be the sentence “Santa is Jolly”
2. Let q be the sentence “ Christmas is on July 4th”
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
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MORE WORDS
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”
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EXAMPLES OF NEGATIONSTATEMENTS
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
4th”
What would be a ~q ?
“Christmas is not on July 4th”
Notice that these negations are also
statements that can be given truth values
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TRUTH TABLES
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
T F
F T
Q ~ Q
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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”
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YOUR TURN TO TRY
Write a negation in standard for each of the statements
1.a: Dallas is the capital of Oklahoma2.b: An orange is a fruit3.c: An apple is not Blue
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CHECK YOUR ANSWERS
1. ~a: Dallas is not the capital of
Oklahoma
2. ~b: An orange is not a fruit
3. ~c: An apple is Blue
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NOW LETS GIVE THE EACH STATEMENT A TRUTH VALUE
a __________~ a _________
c __________~ c _________
b __________
~ b __________
1. a: Dallas is the capital of Oklahoma2. b: An orange is a fruit3. 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
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ANOTHER NEW WORD
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
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EXAMPLES OF CONJUNCTION STATEMENTS
1. Let p be the sentence “Santa is Jolly”
2. Let q be the sentence “ Christmas is on July
4th”
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
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CHECK YOUR ANSWERS
Santa is jolly and Christmas is on July
4th
Santa is jolly and George Washington
was the first U.S. President
Christmas is on July 4th and George
Washington was the first U.S.
President
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WHAT ARE THE TRUTH VALUES?
Santa is jolly and Christmas is on July 4th
FALSE! FALSE! Santa is jolly and George Washington was
the first U.S. President
TRUE! TRUE! TRUE!
Christmas is on July 4th and George
Washington was the first U.S. President
FALSE!!! FALSE!
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A GENERAL TRUTH TABLE FOR P ^ Q
p q p ^ q
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ONE LAST WORD
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
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EXAMPLES OF DISJUNCTION STATEMENTS
1. Let p be the sentence “Santa is Jolly”
2. Let q be the sentence “ Christmas is on July
4th”
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
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CHECK YOUR ANSWERS
Santa is jolly or Christmas is on July 4th
Santa is jolly or George Washington
was the first U.S. President
Christmas is on July 4th or George
Washington was the first U.S.
President
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WHAT ARE THE TRUTH VALUES?
Santa is jolly or Christmas is on July 4th
TRUE! Santa is jolly or George Washington was
the first U.S. President
TRUE! TRUE! TRUE!
Christmas is on July 4th and George
Washington was the first U.S. President
TRUE! TRUE! TRUE!
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WHAT ABOUT………..?????
~ p v q
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A GENERAL TRUTH TABLE FOR P V Q
p q p v q
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NOW TIME TO PUT IT ALL TOGETHER
Given Statements
a: Abraham Lincoln was the 16th U.S.
president
b: A lemon is tart
c: Valentines Day is in December
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QUESTIONS
1. a ^ b2. a v c3. ~a ^ b4. ~ c v ~ a5. 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 ____________