3.5_djj_001_Notes.notebook
October 27, 2009
IN­CLASS TUESDAY, OCT. 27, 2009
Section 3.5: Equivalent Statements, Variations of Conditional Statements, and De Morgan's Laws (Objectives 1­ 5)
Equivalent Statements: statements that have the same truth values in every possible case, symbolized ≡.
Construct a truth table for ~q → ~p.
Recall:
They have the same "answer column". Therefore:
p → q ≡ ~q ’ ~p
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Oct 27­10:41 AM
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Use this equivalent pattern to write a statement equivalent to this conditional: If you live in Tallahassee, then you live in Florida.
"You live in Florida, if you live in Tallahassee."
"You live in Tallahassee only if you live in Florida."
"If you do not live in Florida, then you do not live in Tallahassee."
contrapositive.
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Write the converse, inverse, and contrapositive of the following:
"If it rains, then it pours."
Converse:
__If it pours, then it rains.__(NOT L.E. to p­>q)
Inverse:
__If it does not rain, then it does not pour.
(NOT L.E. to p­>q)
Contrapositive: __If it does not pour, then it does not rain.
Oct 11­12:35 PM
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Write the converse, inverse, and contrapositive of the following:
p → ~ q
Converse:
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Inverse:
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Contrapositive: _____________________________________
Write the converse, inverse, and contrapositive of the following:
"If the cable is out, then we cannot watch TV."
Converse:
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Inverse:
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Contrapositive: _____________________________________
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Are these variations equivalent to the conditional?
Hw.#2a: Use a truth table to show that p → q and ~p ∨ q are equivalent. Oct 11­12:41 PM
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========We got to here on TUE 10­27­09 ===========
Write 2 statements that are equivalent to:
"If the shoe fits, then wear it."
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Write the negation of: October 27, 2009
"If I'm late, then I'll apologize."
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Write the negation of: "If the shoe fits, then wear it."
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Oct 11­12:45 PM
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De Morgan’s Laws
Construct a truth table for ~(p ∧ q).
Construct a truth table for ~p ∨ ~q.
They have the same "answer column". Therefore:
~(p ∧ q) ≡ ~p ∨ ~q
This is one of De Morgan's laws for logic! Oct 11­12:48 PM
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Use this equivalent pattern to write a statement equivalent to: It is not true that Florida and Georgia are both islands.
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Recall from Section 3.2: p: Jon is at work.
q: Mary is at home.
~( p ∧ q ): It is not true that Jon is at work and Mary is at home.
~p ∨ ~q: Jon is not at work and Mary is not at home.
Are they equivalent statements? ________
According to one of De Morgan's laws, what is ~( p ∧ q ) equivalent to? _____________
Oct 11­12:52 PM
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What might De Morgan's other law state is equivalent to ~ (p ∨ q)?
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Use De Morgan's laws to negate ∧ and ∨. Negate: This house is too small or we have too many children.
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Negate: The car is old and it is not reliable.
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Negate: All movies are enjoyable and some movies are dramas.
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Use De Morgan's laws to write a statement that is equivalent to this statement:
"It is not true that I am wrong and you are right."
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Use De Morgan's laws to write a statement that is equivalent to this statement:
"It is not the case that the movie is interesting or entertaining."
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Note: The double negation of a statement is equivalent to the statement. Therefore, ~ (~ p) ≡ p. Use this and De Morgan’s law to derive the disjunction equivalent to the conditional.
~ [~ (p → q)] ≡ ~[p ∧ ~q] ≡ ﴾~p ∨ q)
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It can be tedious to use truth tables to determine whether or not statements are equivalent. Hw. #18: Select the statement that is not equivalent to:
" It is not true that England and Africa are both countries."
a. If England is a country, then Africa is not a country.
b. England is not a country and Africa is not a country.
c. England is not a country or Africa is not a country.
d. If Africa is a country, then England is not a country.
We'll focus on those that can be done using pattern recognition.
Determine which of the statements is equivalent to:
'If the bills are paid, then I can sleep."
a. If the bills are not paid, then I cannot sleep.
b. The bills are not paid or I can sleep.
c. If I can sleep, then the bills are paid.
d. The bills are paid and I cannot sleep.
See text ex. 1­ 9. Use packet pages on Negations and Equivalent Statements.
Oct 11­3:00 PM
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