Notesale: Turn your study into money

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LOGIC
In mathematical and everyday English language, we frequently use logic to express our thoughts verbally
and in writing
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Connecting words such as “or” and “and” and phrases such as “if, then” and “if and only if”
are very common in mathematical definitions, theorems, etc
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CONNECTIVES
OR----The word “OR” is known mathematically as a disjunction and is denoted as ∪ 𝑜𝑟 ∨, both of which
imply the “union” of different propositions typically denoted as P, Q, R, etc
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Note the following example:
Students who have an ACT score of at least 30 OR a GPA of at least 3
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A student who has a 30 or higher on the ACT and a GPA of 3
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A student who has a 30 or higher on the ACT but a GPA less than 3
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A student who has less than a 30 on the ACT but a GPA of at least 3
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The only students who can NOT receive a college scholarship are those who have less than
a 30 on the ACT and a GPA less than 3
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To see this in a truth table format (using 0’s for false and 1’s for true) please note the following:
P
1
1
0
0

Q
1
0
1
0

(P OR Q), (P U Q), (P V Q)
1
1
1
0

XOR---This is known as the “exclusive or” and the difference between this and the previous “or” occurs
in the case where both propositions are true
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In other words, using our example above, a student who has a 30 or higher on the ACT and
a GPA of 3
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That would be a very unhappy student! A better example of
this would be the case where a student is taking a calculus class at 8:00 or a literature class at 8:00
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The student would only take one
class or the other, but not both at 8:00
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The truth table would look as follows:
P
1
1
0
0

Q
1
0
1
0

P⨁Q
0
1
1
0

AND---The word “and” is known mathematically as a conjunction and is denoted as ∩ 𝑜𝑟 ∧, both of
which imply the “intersection” of different propositions
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Note the following example:
Students who have a full-time job and a spouse can receive a housing waiver
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A student who doesn’t
have a full-time job but has a spouse cannot receive a housing waiver
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A student who doesn’t have a
full-time job and doesn’t have spouse cannot receive a housing waiver
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P is called the hypothesis and Q is called the conclusion
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For example, P implies Q, Q only if P, Q whenever P, P → 𝑄, etc
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So, if John is elected as SGA President, then students would be expecting free ice cream
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Students might even still be given ice cream
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We can use the original statement “If P, then Q” to form new conditional statements as follows:
CONVERSE: If Q, then P
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INVERSE: If NOT P, then NOT Q
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” note the following statements:
The converse of our original statement above is If there is a cloud in the sky, then it is raining
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The inverse of our original statement is If it is not raining, then there is not a cloud in the sky
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For this reason, we sometimes use the
contrapositive to rewrite a conditional statement if it is to our benefit
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If it’s true that it’s raining and false that it’s cloudy, you better run for cover!
Biconditional Statements
Biconditional statements are stated as “P if and only if Q”
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Biconditional statements have the same truth table values as “(If P, then Q) AND
(If Q, then P)”
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The truth table
values for a biconditional statement are as follows:
P
1
1
0
0

Q
1
0
1
0

P↔Q
1
0
0
1

Example: You can win the lottery if and only if you purchase a lottery ticket and you have the winning
numbers
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Another way to express “P if and only if Q” is “P is necessary and sufficient for Q”
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If you did not win the lottery then you did not purchase a lottery ticket or you did not purchase a ticket
with the winning numbers indicates the necessity of Q
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For example, 𝑥 + 2 = 5 𝑖𝑓 𝑎𝑛𝑑 𝑜𝑛𝑙𝑦 𝑖𝑓 𝑥 = 3
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Negations
The negation of a proposition such as the hypothesis, P, of a statement is stated as ¬P or sometimes as
~P and is read as “not P” or “it is not the case that P”
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While this is very intuitive for a simple proposition, negations are not always
intuitive for existence or universal statements
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To negate an existence statement, we use the statement “For all such that NOT” or “For every such
that NOT”
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To negate your friend’s claim, you’d need to
show that every student in your calculus class is NOT from Canada
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” The negation of this
statement is “For all x, x is NOT equal to 5
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An example of this in everyday language is if you are trying to argue with a friend who said that every
student in your calculus class has a cellphone
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An example of this in mathematical language is “For every x, y = 2
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”
These quantified statements can also be combined into longer “nested” statements and then negated
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” To negate this statement,
we must examine each quantified part of the statement and apply the negation concepts above
throughout the entire statement
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”
Notice how we negated each of the three following parts of the original statement:
Original
“There exists a y”

Negation
“For every y such that NOT”

“For every nonzero x”

”There exists a nonzero x such that NOT”

“xy = 1”

”xy≠1”

De Morgan’s Laws for negating OR/AND statements:
To negate an “OR” statement, use the following law: ¬(𝑃 𝑜𝑟 𝑄) ≡ ¬𝑃𝑎𝑛𝑑 ¬𝑄
To negate an “AND” statement, use the following law: ¬(𝑃 𝑎𝑛𝑑 𝑄) ≡ ¬𝑃𝑜𝑟 ¬𝑄
These laws make sense in everyday English language as noted in the following examples:
Suppose your friend says that she has a cat or a dog
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In other words, she doesn’t have a cat (not P) AND
she doesn’t have a dog (not Q)
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If you
wanted to negate/refute her claim, you’d only need to show that she either doesn’t have a cat or she
doesn’t have a dog
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Therefore, she doesn’t have both animals as her original claim stated
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The
negation would be that you go to the store with him AND he does NOT buy you a candy bar
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We can now construct truth tables using compound propositions as seen in the following examples:
Example 1: Use truth tables to verify the equivalence of ¬(𝑃 → 𝑄) ≡ 𝑃 ∩ (¬𝑄)
𝑷
1
1
0
0

𝑸
1
0
1
0

𝑷→𝑸
1
0
1
1

¬(𝑷 → 𝑸)
0
1
0
0

¬𝑸
0
1
0
1

𝑷 ∩ ¬𝑸
0
1
0
0

**Notice that the highlighted truth table values in Example 1 are the same
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State the hypothesis and conclusion of the following statements (no claim is made about the
truthfulness of the statements):
a
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b
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c
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d
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e
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f
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g
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2
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3
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4
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**HINT: The contrapositive and the inverse of f
and g may use De Morgan’s Laws
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Construct truth tables for the following compound propositions:
a
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(𝑃 ∪ 𝑄) ∩ (𝑃 ∩ ¬𝑄)
c
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𝑃 ∩ (𝑄 ∩ ¬𝑅)
e
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(𝑃 ∩ 𝑄) ∩ (𝑄 ↔ 𝑅)
g
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Negate the following statements (no claim is made about the truthfulness of the statements):
a
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b
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c
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d
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e
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f
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g
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