MATH1061/7861, Thu 13 Aug 2020
Question 1. Fill in the following table to indicate whether each statement is true or false for each given set S:
an
Neg
ht F
III
t
∀x ∈ S, x is not odd. O
F L
An Inheoations
S = {1,3,5,7} S = {1,2,3,4} S={2,4,6,8}
T
T
∀x ∈ S, x is odd. Tf E
∃x ∈ S such that x is not odd.
Which two statements are negations of each other?
De Morgan gcpfaf Esp EEsq EGD
E
as
Question 2. Negate each of the following statements, and in each case determine whether the original statement is true or
whether the negation is true:
(a) ∃y ∈Rsuch that y2 < 0.
NEI Vyc.IR 570
T
(b) All dogs are white. while NEIizdbheoosdos.gl dTeuoEtwTithet tattoos dis
(c) ∀x ∈ N, either x is prime or x is composite.
Uscis
Gx is cop
T
(d) ∀x ∈ R, if x > 2 then x2 > 4. NEE Face Rst
n
x2 x2
a
3
FocelRst
To
NEI
Fx CIN FxEcN
x is prime nocis prime
Question 3. Determine whether each of the following statements is true or false:
lety xxy TRUE
If O try d y Oy O
(a) ∀x ∈Z, ∃y ∈Zsuch that y = 2x. 5
XII y2 EE 34 z
(b) ∀x ∈Z, ∃y ∈Zsuch that x = 2y. to
TRUE
set no such y.ci FALSE
(c) ∀x ∈ Z, ∃y ∈ Z such that y ≥ x.
Fun
(d) ∃x ∈Zsuch that ∀y ∈Z, x⋅y = x.
x TRUE
Corp
F CEN such that x is neither primenee
L l is notprime
l is not composite
274 and SEE 4
y
composite
0
Question 4. Write down a negation of each of the following statements, and determine whether each statement is true or false:
FEERftp.VyEIR.xty to
Fact Rst HyeIR say TRUE
I
comes after F
is NOTodd a
(a) ∀x ∈R, ∃y ∈Rsuch that x + y = 0. y
(b) ∀x ∈R, ∃y ∈Rsuch that x⋅y = 1. T.IT
(c) ∃x ∈ R such that ∀y ∈ R, x⋅y = 0. FyopeTFfm
TRUE
if They far which say VxetRiFyIRsueuteatx.yfz.f
t i
such that S t Always
Grammy
Question 5. Which one of the following statements is a negation of
“∀x ∈Z, ∃y ∈Zsuch that x + y is odd” ? FxEDL s.t Hye7L say
(a) For every integer x there is an integer y such that x + y is not odd.
(b) There exists an integer x such that for every integer y, x + y is odd.
(c) There exists an integer x such that x + y is even for every integer y.
(d) For every integer x and every integer y there is an odd integer x + y. x
Challenge: Consider the two statements: (a) ∀x ∈ D, ∃y ∈ D such that P(x,y).
(b) ∃y ∈ D such that ∀x ∈ D, P(x,y).
Part 1) Find a domain D and a predicate P(x,y) that makes both statements true. g Part 2) Find a domain D and a predicate P(x,y) that makes both statements false.
Part 3) Find a domain D and a predicate P(x,y) that makes one statements true and one statement false.
Last Thursday’s Challenge 1a. The NAND operation p ⊼ q is defined by p ⊼ q ≡ ∼(p ∧ q). Can you express all of your other logical operations (AND, OR, NOT, IMPLIES) using only NAND and nothing else? Show how to do it, or explain why you cannot.
Last Thursday’s Challenge 1b. Can you express all of your logical operations (AND, OR, NOT, IMPLIES) using only NOT and XOR (⊕)? Show how to do it, or explain why you cannot.
LIMITATIONS of logic
ys ogi
large enough to describe 7L
virus
output eesko
bXa virus Theorem If U is a perfect
virus cheeker
Proof
Godel’s there
gpnb6n
A problems
cannot
ig
that computer programs
incompleteness theorem
that is are
alway statements that we prove to be either true or false
There caurevtsowe.gg
are
cheekiy
input
Program
MB
VIROS Program that modifies the
a virus
operating system
Suppose have V New program
then
itself is puuongfu.tvsTEIhE
bBEn the o S
what happens if give
as input BEh
If
BEN s
3
DEW
is a ui
to BEN om’nated
BEN is a virus
i BEN modified Os
OS when it ran u
i It must have modified
i.visauimst.to Ted