Material from beginning of semester through quantifiers (e.g., statements and
non-statements, conditional and biconditional statements, propositional logic,
quantifiers, etc.)
3 - 5 short-answer questions, possibly a longer question where I ask you to
write a formal proof
Whole class period
Open book, notes, computer; closed person
Extended grading deadline for problem set 4: grade by Friday Sept. 30
Questions?
Quantifiers
Section 2.4
Rephrase in English or symbolically, and decide whether true or false
English phrases: ∃ means “some” ∀ means
“every,” “all”
“Some” and “all” indicate which values from universal
set to apply to predicate
Some natural number is a multiple of 9
(∃ x∈ℕ) ( (∃ y∈ℤ) y = x/9 )
Or (∃ x∈ℕ) ( (∃ y∈ℤ) x = 9y )
Every natural number is a multiple of 9
(∀ x∈ℕ) ( (∃ y∈ℤ) y = x/9 )
For every real number x, x2 - 1 = (x+1)(x-1)
(∀x∈ℝ) ( x2 - 1 = (x+1)(x-1) )
(∃n∈ℕ) (n = 32)
Some natural number n = 32
(∀x∈{5,7,9}) (x > 3)
For all elements x in {5,7,9}, x is greater than 3
(∃a∈ℝ) (∀x∈ℝ) (ax = x)
There is some real number a such that for every real number x, ax = x
Rephrase using negation equivalencies
Switch “exists” for “for all” and negate predicate
¬(∀x ∈ U) P(x) ≡ (∃x∈U) (¬P(x))
¬(∃x∈U) P(x) ≡ (∀x∈U) (¬P(x))
Not all that is gold glitters
Some gold does not glitter
There is no real number whose square is less than 0
All real numbers have squares greater than or equal to 0
Do the equivalences in Theorem 2.16 (negations of quantified statements) look
familiar?