Question

**q1**, Let A, B be sets such that A ∩ B = B. Show
that B − A = ∅

**q2,** (∀x ∈ N)(∀y ∈ N). (xy = 0 ⇒ (x = 0 ∧ y =
0)) It is true or not

Thanks

Answer #1

Let Q1, Q2, Q3 be constants so that (Q1, Q2) is the critical
point of the function f(x, y) = xy + y − x, and Q3 = 1 if f has a
local minimum at (Q1, Q2), Q3 = 2 if f has a local maximum at (Q1,
Q2), Q3 = 3 if f has a saddle point at (Q1, Q2), and Q3 = 4
otherwise. Let Q = ln(3 + |Q1| + 2|Q2| + 3|Q3|). Then T =...

Let Q1, Q2, Q3 be constants so that (Q1, Q2) is the critical
point of the function f(x, y) = xy − 5x − 5y + 25, and Q3 = 1 if f
has a local minimum at (Q1, Q2), Q3 = 2 if f has a local maximum at
(Q1, Q2), Q3 = 3 if f has a saddle point at (Q1, Q2), and Q3 = 4
otherwise. Let Q = ln(3 + |Q1| + 2|Q2| + 3|Q3|). Then...

(a) Let A and B be countably infinite sets. Decide whether the
following are true for all, some (but not all), or no such sets,
and give reasons for your answers. A ∪B is countably infinite A
∩B is countably infinite A\B is countably infinite, where A ∖ B =
{ x | x ∈ A ∧ X ∉ B }. (b) Let F be the set of all total unary
functions f : N → N...

2. Let Q1 = y(2), Q2 = y(3), where y = y(x) solves y' + 2xy =
2x^3 , y(0) = 1. Let Q = ln(3 + |Q1| + 2|Q2|). Then T = 5 sin^2
(100Q) satisfies:— (A) 0 ≤ T < 1. — (B) 1 ≤ T < 2. — (C) 2 ≤
T < 3. — (D) 3 ≤ T < 4. — (E) 4 ≤ T ≤ 5.

Let A, B be sets and f: A -> B. For any subsets X,Y subset of
A, X is a subset of Y iff f(x) is a subset of f(Y).
Prove your answer. If the statement is false indicate an
additional hypothesis the would make the statement true.

Let A, B, C be three sets. Show that A ∪ B = A ∩ C ⇐⇒ B ⊆ A ⊆
C.

Let Q1, Q2, Q3, Q4 be constants so that y =Q1+Q2x+Q3x^2+Q4x^3
satisfies that y(1)=1 and (1-x^2)y"-2xy'+12y=0.

· Let A and B be sets. If A and B are countable, then A ∪ B is
countable.
· Let A and B be sets. If A and B are infinite, then A ∪ B is
infinite.
· Let A and B be sets. If A and B are countably infinite, then A
∪ B is countably infinite.
Find nontrivial sets A and B such that A ∪ B = Z, then use these
theorems to show Z is...

Let X, Y and Z be sets. Let f : X → Y and g : Y → Z functions.
(a) (3 Pts.) Show that if g ◦ f is an injective function, then f is
an injective function. (b) (2 Pts.) Find examples of sets X, Y and
Z and functions f : X → Y and g : Y → Z such that g ◦ f is
injective but g is not injective. (c) (3 Pts.) Show that...

Let Q1=y(1.5), Q2=y(2), where y=y(x) solves y'+ycotx=y^3 sin^3x
y( π/2)=1
Please show all steps!
Thank you!

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