Other Math Answers

Set up a complete truth table for each of the following compound statements. Write your answers on a separate sheet of paper.

1. ~(~P) 4. ~P Λ ~Q 7. ~(~P Λ ~Q) 10. ~(~P Λ ~P)

2. P Λ ~Q 5. ~[~ (~P)] 8. P Λ ~P 11. (P Λ Q) Λ R

3. ~(P Λ Q) 6. ~(P Λ ~Q) 9. ~P Λ P 12. P Λ (Q Λ R)

(Military Time) Evaluate each of the following, where ⊕ and ⊖ indicate addition and subtraction, respectively, on a 24-hour clock.

1. 1800   +   0900
2. 0800   +   0900
3. 1600   +   1200
4. 1300   +   1300
5. 1000   +   1400
6. 1200   –   0700
7. 1800   –   1900
8. 0200   –   0500
9. 0900   –   1800
10.  0600   –   2200

1. Let a and b be two cardinal numbers. Modify Cantor’s definition of a < b to define a ≤ b. (Hint: Examine what happens if you drop condition (a) from Cantor’s definition of a < b.) 2. Prove that a ≤ a. 3. Prove that if a ≤ b and b ≤ c, then a ≤ c. 4. Do you think that a ≤ b and b ≤ a imply

a = b? Explain your reasoning. (Hint: This is not as trivial as it might look.)

Let f : A → B be a function. Show that the following two conditions are equivalent: 1. f is one-to-one.

2. For each a1, a2 ∈ A, whenever f(a1) = f(a2), then a1 = a2.

Let f : A → B be a function.

1. Show that for the identity function iA on A we have f ◦ iA = f.

2. Show that for the identity function iB on B we have iB ◦ f = f.

Let f : A → B, g : B → C, and h : C → D be functions.

1. State what you need to show to conclude that h ◦ (g ◦ f) = (h ◦ g) ◦ f. 13

2. Consider now some a ∈ A. Calculate h((g ◦ f)(a)) and (h ◦ g)(f(a)). Are they equal?

3. Use your solutions to (1)–(2) to conclude that h ◦ (g ◦ f) = (h ◦ g) ◦ f.

a) Difference between area integral and surface integral. (2)

b) Find the area of the region enclosed by the circle 𝑥

2 + 𝑦

2 = 4.

(4)

c) Find the area of the region enclosed by the ellipse 𝑥

2

9

+

𝑦

2

4

= 1.

Let f : A → B be a function. Show that the following two conditions are equivalent: 1. f is one-to-one.

2. For each a1, a2 ∈ A, whenever f(a1) = f(a2), then a1 = a2.

A student organization has money left over in its budget and must spend it before the school year ends. The

members arrived at five different possibilities:

Rank

Establish a scholarship 1 2 3 3 4

Pay for some members to attend a convention 2 1 2 1 5

Buy new computers 3 3 1 4 1

Throw a year-end party 4 5 5 2 2

Donate to charity 5 4 4 5 3

Number of votes 8 5 12 9 7

Determine how the money should be spent using

(a) winners-runoff

(b) losers-eliminated

(c) Use Borda count