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**Kryptonis**- Replies: 1

Could use a hint on how to begin this problem:

∫cos^-1 (x) dx

Thanx a bunch, was totally going in the wrong direction on this

**Kryptonis**- Replies: 10

I'm having a bit of a time with this. I have tried integration by parts and just can't seem to get it right....

∫(3-3x)/(sqrt(64-9x^2)) dx

**Kryptonis**- Replies: 2

x = t / (3+t) y = ln(3 + t) 0 ≤ t ≤ 4

Here's the problem, my real question lies in the answer...

-sqrt(u^2 + 9)/(u) + ln(u + sqrt(u^2 + 9)) with the lower @ 3 and the upper @ 7

where u = t + 3 and du = dt

When I integrate, I'm at a loss as to how i should be coming up with -sqrt(u^2 + 9)/(u) + ln(u + sqrt(u^2 + 9)).

As I keep coming up with -sqrt(u^2 + 9)/(u) + ln(sqrt(u^2 + 9).

Ty, got this one shortly after i posted. Thanx for the help though. Much appreciated!

ty sir, much appreciated!

A − B ⊆ C ≡ (A ∩ ¬B) ⊆ C Def of Diff

≡ {x | (x ∈ A ∧ x ¬∈ B) → x ∈ C} Def of Diff, Def of subset

≡ {x | (x ¬∈ A ∨ x ∈ B) ∨ x ∈ C} Log Equiv, De Morg

≡ {x | (x ∈ A → x ∈ B) ∪ x ∈ C} Log Equiv, Def of Union

≡ A ⊆ B ∪C Set Builder Notation

**Kryptonis**- Replies: 1

Let A, B and C be arbitrary sets taken from the positive integers.

Prove the following statement: If A − B ⊆ C , then A ⊆ B ∪C

**Kryptonis**- Replies: 3

a) Determine which of the followings are functions with domain X.

i) (3 pts) X = {1, 3, 5, 7, 8} and R ={(1,7), (3,5), (5,3), (7, 7), (8,5)}

ii) (3 pts) X = {-2, -1, 0, 1} and R = {(-2, 6), (0, 3), (1, -1)}

iii) (3 pts) X is the set of real numbers and, for x ∈ X,

g(x) = x^2 − 3x + 2, assume that the codomain is also X

iv) (3 pts) X is the set of real numbers and, for x ∈ X,

g(x) = sqrt(x^2 − 3x + 2) , assume that the codomain is also X

v) (3 pts) X is the set of real numbers and, for x ∈ X, g(x) = log2 x , , assume that the codomain is also X

b) Let Z = {...−2, −1, 0, 1, 2, ...} denote the set of integers. Suppose f : Z→Z is a function, defined by:

f (n) = {2 if is odd

n/ 2 if is even

i) (5pts) Prove or disprove that f is one-to-one (injective)

ii) (5pts) Prove or disprove that f is onto (surjective).

**Kryptonis**- Replies: 2

Prove the following assertions for sets A and B from an universe U without using Venn Diagrams or membership tables:

a) (10 pts) A ⊆ B if and only if A ∩ ¬B = ∅.

b) (10 pts) A ⊆ B if and only if ¬A ∪ B = U.

**Kryptonis**- Replies: 1

Let A, B, and C be sets such that C ⊂ B (i.e., C is a proper subset of B, or possibly C = B). Use appropriate set theoretic laws and theorems to prove that (A B) ∪ (B C) = ¬C ∩ (A ∪ B). Be sure to explain each step of your proof.

This is what i have, and i have tried several ways just can't quite seem to get it right... Any help would be great, ty in advance!

(A B) ∪ (B C)

≡{x | x ∈ A ∧ x B}∪{x | x ∈ B ∧ x C} Def of diff

≡{x | (x ∈ A ∧ x B) ∨ ( x ∈ B ∧ x C)} Def of union

≡{x | (x ∈ A ∧ x B) ∨ ( x B ∨ x ∈ C)} De Morgan

≡{x | x ∈ A ∧ (x B ∨ x ∈ C)} Idem & Assoc

≡{x | x ∈ A ∧ (x ∈ B ∧ x C)} De Morgan

≡{x | x C x ∧ (∈ A ∧ x ∈ B)} Assoc

≡{x | x C x ∩ (∈ A ∩ x ∈ B)} Def of Inter

≡ ¬C ∩ (A ∩ B) Def of Set Build Notation

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