Question

1. Write the following sets in list form. (For example, {x | x ∈N,1 ≤ x < 6} would be {1,2,3,4,5}.) (a) {a | a ∈Z,a2 ≤ 1}. (b) {b2 | b ∈Z,−2 ≤ b ≤ 2} (c) {c | c2 −4c−5 = 0}. (d) {d | d ∈R,d2 < 0}.

2. Let S be the set {1,2,{1,3},{2}}. Answer true or false: (a) 1 ∈ S. (b) {2}⊆ S. (c) 3 ∈ S. (d) {1,3}∈ S. (e) {1,2}∈ S (f) {2}∈ S. (g) {2,{2}}⊆ S. (h) {3}⊆ S. (i) ∅∈ S. (j) ∅⊆ S.

3. Let A = N, B = {y3 | y ∈Z}, C = {z + 3 | z ∈Z, −2 ≤ z ≤ 10}, D = Q. For each of the following, state whether they are true or false. If true, give a reason; if false, give a counterexample. (a) A ⊆ B. (b) B ⊆ A. (c) C ⊆ A. (d) A∪B ⊆ D. (e) C ∩D ⊆ B.

4. For each of the following, give an example of a statment P and a statement Q that satisfy the given conditions. (a) P ⇒ Q but Q 6⇒ P. (b) P if and only if Q. (c) P if Q.

5. Prove that if x is an odd integer, then there exists some
integer y such that x2 = 4y +1

Answer #1

Consider p(x) and q(x), where x ∈ U = {1, 2}. If the following
is true, give a rigorous argument. If it is false, give a
counterexample. (Note that “p implies q” is the same as “if p, then
q” and also as “p → q.”) (i) (∀x ∈ U, p(x) → q(x)) implies [ (∀x ∈
U, p(x)) → (∀x ∈ U, q(x)) ] ? What about its converse ? (ii) (∃x ∈
U, p(x) → q(x)) implies [...

1. For each statement that is true, give a proof and for each
false statement, give a counterexample
(a) For all natural numbers n, n2
+n + 17 is prime.
(b) p Þ q and ~ p Þ ~ q are NOT logically
equivalent.
(c) For every real number x
³ 1, x2£
x3.
(d) No rational number x satisfies
x^4+ 1/x
-(x+1)^(1/2)=0.
(e) There do not exist irrational numbers
x and y such that...

Suppose K is a nonempty compact subset of a metric space X and
x∈X.
Show, there is a nearest point p∈K to x; that is, there
is a point p∈K such that, for all other q∈K,
d(p,x)≤d(q,x).
[Suggestion: As a start, let S={d(x,y):y∈K} and show there is a
sequence (qn) from K such that the numerical sequence (d(x,qn))
converges to inf(S).] Let X=R^2 and T={(x,y):x^2+y^2=1}.
Show, there is a point z∈X and distinct points a,b∈T
that are nearest points to...

Consider the function F(x, y, z) =x2/2−
y3/3 + z6/6 − 1.
(a) Find the gradient vector ∇F.
(b) Find a scalar equation and a vector parametric form for the
tangent plane to the surface F(x, y, z) = 0 at the point (1, −1,
1).
(c) Let x = s + t, y = st and z = et^2 . Use the multivariable
chain rule to find ∂F/∂s . Write your answer in terms of s and
t.

Consider plane P: 4x -y + 2z = 8, line: <x, y, z> =
<1+t, -1+2t, 3t>, and point Q(2,-1,3)
b) Find the perpendicular distance between point Q and plane
P

Q(x,y) is a propositional function and the domain for the
variables x & y is: {1,2,3}.
Assume Q(1,3), Q(2,1), Q(2,2), Q(2,3), Q(3,1), Q(3,2) are true,
and Q(x,y) is false otherwise.
Find which statements are true.
1. ∀yƎx(Q(x,y)->Q(y,x))
2. ¬(ƎxƎy(Q(x,y)/\¬Q(y,x)))
3. ∀yƎx(Q(x,y) /\ y>=x)

For each set of conditions below, give an example of a predicate
P(n) deﬁned on N that satisfy those conditions (and justify your
example), or explain why such a predicate cannot exist.
(a) P(n) is True for n ≤ 5 and n = 8; False for all other
natural numbers.
(b) P(1) is False, and (∀k ≥ 1)(P(k) ⇒ P(k + 1)) is True.
(c) P(1) and P(2) are True, but [(∀k ≥ 3)(P(k) ⇒ P(k + 1))] is
False....

(1) Determine whether the propositions p → (q ∨ ¬r) and (p ∧ ¬q)
→ ¬r are logically equivalent using either a truth table or laws of
logic.
(2) Let A, B and C be sets. If a is the proposition “x ∈ A”, b
is the proposition “x ∈ B” and
c is the proposition “x ∈ C”, write down a proposition involving a,
b and c that is logically equivalentto“x∈A∪(B−C)”.
(3) Consider the statement ∀x∃y¬P(x,y). Write down a...

Let A, B, C, D be sets, and consider the following:
Theorem 1. A × (B ∪ C) = (A × B) ∪ (A × C).
Theorem 2. (A × B) ∩ (C × D) = (A ∩ C) × (B ∩ D).
Theorem 3. (A × B) ∆ (C × D) = (A ∆ C) × (B ∆ D).
For each, give a proof or counterexample.

1. Which of the following sets in (a) are groups under addition?
For each set which is not a group under addition, show which group
property does not apply by counterexample.
a. N; W; Z; Q; R; E; C; P(x, 3); M(2,1,N) .

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