L05 - Negating Statements
CSci/Math 2112
15 May 2015
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Assignment 1
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Assignment 1 is now posted
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Due May 22 at the beginning of class
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Can work on it in groups, but separate write-up
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Don’t forget your name, B00# on first page; staple
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Questions in the right order
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Points for each question are in the margin
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More on Quantifiers and Implications
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Instead of ∃! you can always use ∃, it only emphasizes that
there is exactly one (similar to ( instead of ⊆)
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More on Quantifiers and Implications
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Instead of ∃! you can always use ∃, it only emphasizes that
there is exactly one (similar to ( instead of ⊆)
Unless asked to, do not use symbolic shorthand on
assignments or tests! Use full sentences.
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More on Quantifiers and Implications
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Instead of ∃! you can always use ∃, it only emphasizes that
there is exactly one (similar to ( instead of ⊆)
Unless asked to, do not use symbolic shorthand on
assignments or tests! Use full sentences.
I
Symbolic shorthand is very useful though for scrap work!
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More on Quantifiers and Implications
I
I
Instead of ∃! you can always use ∃, it only emphasizes that
there is exactly one (similar to ( instead of ⊆)
Unless asked to, do not use symbolic shorthand on
assignments or tests! Use full sentences.
I
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Symbolic shorthand is very useful though for scrap work!
For some implications P ⇒ Q the truth value depends on a
variable x. These are interpreted as ∀x ∈ A, P ⇒ Q.
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Pop Quiz!
Don’t forget to put your name on your quiz!
Question 1
According to DeMorgan’s Laws, we have ∼ (P ∧ Q) = . . .
Question 2
The only case in which P ⇒ Q is false is if . . . Therefore
∼ (P ⇒ Q) = . . .
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English to Symbolic
Questions about Section 2.9 (Translating English to Symbolic
Language)?
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Negations
Why do we need to know how to negate a statement?
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The contrapositive of an implication is ∼ Q ⇒∼ P.
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Negations
Why do we need to know how to negate a statement?
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The contrapositive of an implication is ∼ Q ⇒∼ P.
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To find examples showing that a statement is false (a
counterexample).
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Negations
Why do we need to know how to negate a statement?
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The contrapositive of an implication is ∼ Q ⇒∼ P.
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To find examples showing that a statement is false (a
counterexample).
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P ⇒ Q =∼ P ∨ Q
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Negations
Why do we need to know how to negate a statement?
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The contrapositive of an implication is ∼ Q ⇒∼ P.
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To find examples showing that a statement is false (a
counterexample).
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P ⇒ Q =∼ P ∨ Q
A lot of mistakes are made negating statements - pay attention to
details and practice!
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Negation: Truth Tables
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One way to negate a statement is to look at its truth table
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Negation: Truth Tables
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One way to negate a statement is to look at its truth table
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Flip all entries
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Negation: Truth Tables
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One way to negate a statement is to look at its truth table
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Flip all entries
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Do the disjunctive normal form of the negated statement
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Negations: And, Or, Xor
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∼ (P ∨ Q) =∼ P∧ ∼ Q (DeMorgan’s Law)
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∼ (P ∧ Q) =∼ P∨ ∼ Q (DeMorgan’s Law)
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∼ (P ⊕ Q) = (P ∧ Q) ∨ (∼ P∧ ∼ Q)
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Negations: And, Or, Xor
Example 1
Negate each of the following statements.
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Negations: And, Or, Xor
Example 1
Negate each of the following statements.
(a) I like reading and playing games.
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Negations: And, Or, Xor
Example 1
Negate each of the following statements.
(a) I like reading and playing games.
(b) Either Pearl or Bob was in the ballroom, but not both.
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Negations: And, Or, Xor
Example 1
Negate each of the following statements.
(a) I like reading and playing games.
(b) Either Pearl or Bob was in the ballroom, but not both.
(c) I don’t like being sick or tired.
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Negations: And, Or, Xor
Example 1
Negate each of the following statements.
(a) I like reading and playing games.
(b) Either Pearl or Bob was in the ballroom, but not both.
(c) I don’t like being sick or tired.
(d) The integer x is divisible by 2 and by 3.
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Negations: And, Or, Xor
Example 1
Negate each of the following statements.
(a) I like reading and playing games.
(b) Either Pearl or Bob was in the ballroom, but not both.
(c) I don’t like being sick or tired.
(d) The integer x is divisible by 2 and by 3.
(e) At least one of the integers x and y is even.
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Negation: Quantifiers
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∼ ∃ - there is not a single one satisfying the statement, thus
for all (∀) the statement is false
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Negation: Quantifiers
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∼ ∃ - there is not a single one satisfying the statement, thus
for all (∀) the statement is false
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∼ (∃x ∈ S, P(x)) = ∀x ∈ S, ∼ P(x)
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Negation: Quantifiers
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∼ ∃ - there is not a single one satisfying the statement, thus
for all (∀) the statement is false
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∼ (∃x ∈ S, P(x)) = ∀x ∈ S, ∼ P(x)
∼ ∀ - there is at least one (∃) for which the statement isn’t
true
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Negation: Quantifiers
I
∼ ∃ - there is not a single one satisfying the statement, thus
for all (∀) the statement is false
I
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∼ (∃x ∈ S, P(x)) = ∀x ∈ S, ∼ P(x)
∼ ∀ - there is at least one (∃) for which the statement isn’t
true
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∼ (∀x ∈ S, P(x)) = ∃x ∈ S, ∼ P(x)
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Negation: Quantifiers
Example 2
Negate the following:
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Negation: Quantifiers
Example 2
Negate the following:
(a) Some subsets of N have 5 elements.
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Negation: Quantifiers
Example 2
Negate the following:
(a) Some subsets of N have 5 elements.
(b) For every real number x there is a real number y for which
y 3 = x.
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Negation: Quantifiers
Example 2
Negate the following:
(a) Some subsets of N have 5 elements.
(b) For every real number x there is a real number y for which
y 3 = x.
(c) For any two integers x and y ,
x
y
is a rational number.
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Negation: Quantifiers
Example 2
Negate the following:
(a) Some subsets of N have 5 elements.
(b) For every real number x there is a real number y for which
y 3 = x.
(c) For any two integers x and y ,
x
y
is a rational number.
(d) There is an integer a such that for all integers b we have
a + b = b.
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Negation: Quantifiers
Example 2
Negate the following:
(a) Some subsets of N have 5 elements.
(b) For every real number x there is a real number y for which
y 3 = x.
(c) For any two integers x and y ,
x
y
is a rational number.
(d) There is an integer a such that for all integers b we have
a + b = b.
(e) For every integer a there exists an integer b such that
a + b = 0.
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Negation: Implication
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∼ (P ⇒ Q) = P∧ ∼ Q
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Negation: Implication
Example 3
Negate the following:
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Negation: Implication
Example 3
Negate the following:
(a) If Pearl was in the ballroom, then so was Venom.
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Negation: Implication
Example 3
Negate the following:
(a) If Pearl was in the ballroom, then so was Venom.
(b) If a number is divisible by 6, then it is even.
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Negation: Implication
Example 3
Negate the following:
(a) If Pearl was in the ballroom, then so was Venom.
(b) If a number is divisible by 6, then it is even.
(c) If you came to class today, then you were up before 10:30am.
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Negation: Implication
Example 3
Negate the following:
(a) If Pearl was in the ballroom, then so was Venom.
(b) If a number is divisible by 6, then it is even.
(c) If you came to class today, then you were up before 10:30am.
(d) If x ∈ Z, then −x ∈ Z.
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Negation: Everything combined
Example 4
Negate the following:
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Negation: Everything combined
Example 4
Negate the following:
(a) If the street is wet, then it rained or there was another water
source.
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Negation: Everything combined
Example 4
Negate the following:
(a) If the street is wet, then it rained or there was another water
source.
(b) You came to class an you were either on time or late, but not
both.
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Negation: Everything combined
Example 4
Negate the following:
(a) If the street is wet, then it rained or there was another water
source.
(b) You came to class an you were either on time or late, but not
both.
(c) If for all x ∈ A we know that 2x ∈ A as well, then either A is
empty, or A is infinite, but not both.
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Negation: Everything combined
Example 4
Negate the following:
(a) If the street is wet, then it rained or there was another water
source.
(b) You came to class an you were either on time or late, but not
both.
(c) If for all x ∈ A we know that 2x ∈ A as well, then either A is
empty, or A is infinite, but not both.
(d) If A = Z, then for all x ∈ A we know that 2x ∈ A as well.
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Negation: Everything combined
Example 4
Negate the following:
(a) If the street is wet, then it rained or there was another water
source.
(b) You came to class an you were either on time or late, but not
both.
(c) If for all x ∈ A we know that 2x ∈ A as well, then either A is
empty, or A is infinite, but not both.
(d) If A = Z, then for all x ∈ A we know that 2x ∈ A as well.
(e) Every integer that is not odd is even.
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