Chapter 8 Logic DP Studies. Content A Propositions B Compound propositions C Truth tables and...
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Chapter 8 Logic DP Studies. Content A Propositions B Compound propositions C Truth tables and logical equivalence D Implication and equivalence E Converse,
Content A Propositions B Compound propositions C Truth tables
and logical equivalence D Implication and equivalence E Converse,
inverse, and contrapositive F Valid arguments b
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Opening Problem b
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A. Propositions Propositions are statements which may be true
or false. Examples: 1. The sky is green. 2. There is a virus on
this computer.
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A. Propositions Questions are not propositions. Is the sky
green today? Comments or opinions that are subjective, are also not
propositions since they are not definitely true or false. Green is
a nice color Propositions may be indeterminate. your father is 50
years old would not have the same answer (true or false) for all
people.
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A. Proposition The truth value of a proposition is whether it
is true or false. example: 1. one plus one equals two ( T ) 2. one
plus one equals three ( F )
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A. Proposition Example 1
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A. Proposition Solution to example 1
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A. Proposition We represent propositions by letters such as p,
q, and r. For example, p: It always rains on Tuesdays. q: 37 + 9 =
46 r: x is an even number.
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A. Proposition
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We can also represent negation using a truth table. p TF
FT
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A. Proposition Example 2
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A. Proposition Solutions to example 2:
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A. Proposition We can use a Venn diagram to represent these
propositions and their negations.
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A. Proposition Example 3
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A. Proposition Solutions to example 3:
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B. Compound Propositions Compound propositions are statements
which are formed using connectives such as and and or. For example:
p: Bob went to the beach q: Bob flew a kite p ^ q: Bob went to the
beach and flew a kite
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B. Compound Propositions When two propositions are joined using
the word and, the new proposition is the conjunction of the
original propositions. If p and q are propositions, p ^ q is used
to denote their conjunction. For example: p: Eli had soup for lunch
q: Eli had a pie for lunch p ^ q: Eli had soup and a pie for lunch.
A conjunction is true only when both original propositions are
true.
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B. Compound Propositions Conjunction truth table pqp ^ q TTT
TFF FTF FFF p ^ q is true when both p and q are true. The first 2
columns list the possible combinations for p and q.. p ^ q is false
whenever one or both of p and q are false.
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B. Compound Propositions
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When two propositions are joined by the word or, the new
proposition is the disjunction of the original propositions. If p
and q are propositions, p V q is used to denote their disjunction.
For example: p: Frank played tennis today q: Frank played golf
today p V q: Frank played tennis or golf today. p V q is true if
Frank played tennis or golf or both today. A disjunction is true
when one or both propositions are true. A disjunction is only false
if both propositions are false.
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B. Compound Propositions The truth table for the disjunction p
or q is: pqp V q TTT TFT FTT FFF p V q is true if p or q or both
are true. p V q is only false if both p and q are false.
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B. Compound Propositions We can use Venn diagrams to represent
disjunction.
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B. Compound Propositions The exclusive disjunction is true when
only one of the propositions is true. The exclusive disjunction of
p and q is written p V q. We can describe p V q as p or q, but not
both, or exactly one of p and q. For example: p: Sally ate cereal
for breakfast q: Sally ate toast for breakfast p V q: Sally ate
cereal or toast, but not both, for breakfast.
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B. Compound Propositions The truth table for the exclusive
disjunction p V q is: pqp V q TTF TFT FTT FFF p Y q is true if
exactly one of p and q is true. p Y q is false if p and q are both
true or both false.
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B. Compound Propositions If P and Q are the truth sets for
propositions p and q respectively, then the truth set for p V q is
the region shaded, where exactly one of p and q is true.
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B. Compound Propositions Example 4:
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B. Compound Propositions Solutions to example 4:
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C. Truth Tables and Logical Equivalence This is what we know so
far.
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C. Truth Tables and Logical Equivalence
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Solution to example 5
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C. Truth Tables and Logical Equivalence Solution to example
5
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C. Truth Tables and Logical Equivalence Solution to example
5
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C. Truth Tables and Logical Equivalence A compound proposition
is a tautology if all the values in its truth table column are
true. Example:
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C. Truth Tables and Logical Equivalence
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Solution to example 8:
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C. Truth Tables and Logical Equivalence When three propositions
are under consideration, we usually denote them p, q, and r. The
possible combinations of the truth values for p, q, and r are
listed systematically in the table alongside.
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C. Truth Tables and Logical Equivalence Example 9: Construct a
truth table for the compound proposition (p V q) r.
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C. Truth Tables and Logical Equivalence Solution to example
9:
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D. Implication and Equivalence If two propositions can be
linked with If.... then...., then we have an implication. The
implicative statement if p then q is written p => q and reads p
implies q. p is called the antecedent and q is called the
consequent. For example: Given p: Kato has a TV set, and q: Kato
can watch TV, we have p => q: If Kato has a TV set, then Kato
can watch TV.
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C. Truth Tables and Logical Equivalence the truth table for p
=> q is:
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C. Truth Tables and Logical Equivalence If two propositions are
linked with .... if and only if...., then we have an equivalence.
The equivalence p if and only if q is written p q. p q is logically
equivalent to the conjunction of the implications p => q and q
=> p. Example: Consider p: I will pass the exam, and q: The exam
is easy. We have p => q: If I pass the exam, then the exam is
easy. q => p: If the exam is easy, then I will pass it. p q: I
will pass the exam if and only if the exam is easy.
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C. Truth Tables and Logical Equivalence We can find the truth
table for p, q by constructing the truth table of its logical
equivalent (p => q) (q => p) So, the truth table for
equivalence p q is
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E. Converse, Inverse, and Contrapositive The converse of the
statement p => q is the statement q => p. Example: p => q:
If it is a red car, then its a convertible q => p: If its a
convertible, then it is a red car The truth table for converse
is
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E. Converse, Inverse, and Contrapositive Example 10: For p: the
triangle is isosceles, and q: two angles of the triangle are equal,
state p => q and its converse q => p.
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E. Converse, Inverse, and Contrapositive Solution to example
10: p => q: If the triangle is isosceles, then two of its angles
are equal. q => p: If two angles of the triangle are equal, then
the triangle is isosceles
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E. Converse, Inverse, and Contrapositive
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the converse and inverse of an implication are logically
equivalent. converse inverse
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E. Converse, Inverse, and Contrapositive
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p => q
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E. Converse, Inverse, and Contrapositive For example, consider
p: Sam is in the library and q: Sam is reading.
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E. Converse, Inverse, and Contrapositive Example 11: Write down
the contrapositive of: All teachers drive blue cars.
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E. Converse, Inverse, and Contrapositive Solution to example
11:
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F. Valid Arguments An argument is made up of a set of
propositions, called the premise, that leads to a conclusion. An
argument is usually indicated by a word such as therefore or hence.
Example: If George is at the beach, then he is getting sunburnt.
George is at the beach. Therefore, George is getting sunburnt.
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E. Converse, Inverse, and Contrapositive We set out arguments
by separating the premise and the conclusion with a horizontal
line. If George is at the beach, then he is getting sunburnt.
George is at the beach. George is getting sunburnt. premise
conclusion
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E. Converse, Inverse, and Contrapositive So, from the
propositions p => q and p, we are implying the conclusion q. We
can write this argument in logical form as (p => q) p => q.
To determine whether this argument is valid, we construct a truth
table for this proposition, and see whether it is a tautology. We
have a tautology, so our argument is valid. The conclusion we have
made follows logically from the premise.
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E. Converse, Inverse, and Contrapositive Example 12: Determine
the validity of the following argument: If a triangle has three
sides, then 2 +4 = 7. 2 + 4 = 7 Hence, a triangle has three
sides.
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E. Converse, Inverse, and Contrapositive Solution to example
12:
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E. Converse, Inverse, and Contrapositive IMPORTANT: Proposition
q is clearly false. However, this does not affect the validity of
the argument. Logic is not concerned with whether the premise is
true or false, but rather with what can be validly concluded from
the premise. So if we started out with a wrong premise, our
conclusion is wrong, but it is logically sound.
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E. Converse, Inverse, and Contrapositive Example 13: Determine
the validity of the following argument: If x is a natural number,
then x is an integer. If x is an integer, then x is rational.
Therefore, if x is a natural number, then x is rational.
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E. Converse, Inverse, and Contrapositive Solution to example
13: