Let A, B, and C be sets and suppose $A\subseteq B$, $B\subseteq C$, and $C\subseteq A$. Prove that A=C.

Let A, B, and C be sets. Show that a) (A ∪ B) ⊆ (A ∪ B ∪ C). b) (A ∩ B ∩ C) ⊆ (A ∩ B). c) (A − B) − C ⊆ A − C. d) (A − C) ∩ (C − B) = ∅. e) (B − A) ∪ (C − A) = (B ∪ C) − A.

Use the functions to write an equation for $f(x_k)$.

$$f(x)=d(x-2)^{2}$$

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$$\begin{matrix}\text{k} & \mathrm{x_k}\\\text{1} & \text{2.5}\\\text{2} & \text{3.0}\\\text{3} & \text{3.5}\\\text{4} & \text{4.0}\\\end{matrix}$$

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The Barron’s Big Money Poll asked 131 investment managers across the United States about their short-term investment outlook (Barron’s, October 28, 2002). Their responses showed 4% were very bullish, 39% were bullish, 29% were neutral, 21% were bearish, and 7% were very bearish. Let x be the random variable reflecting the level of optimism about the market. Set x = 5 for very bullish down through x = 1 for very bearish.

Develop a probability distribution for the level of optimism of investment managers.

Write each of the following sets in set-builder notation. $\{3,6,11,18,27,38, \ldots\}$

Evaluate the determinant.

$$\left| \begin{array} { r r } { - 1 } & { 2 } \\ { 3 } & { - 4 } \end{array} \right|$$

The double factorial n!! is defined for odd positive integers n; it is the product of all odd numbers from 1 to n inclusive. For example, 7!!=1$\times$3$\times$5$\times$7=105. Please answer the following. a. Evaluate 9!!. b. For an odd integer n, are n!! and (n!)! equal? c. Write an expression for n!! using product notation. d. Explain why this formula works:

$$(2k-1)!!=\frac{(2k)!}{k!2^k}$$

Add or subtract. Identify any x-values for which the expression is undefined.

$$\frac { 3 x + 2 } { x - 2 } - \frac { x + 5 } { x - 2 }$$

Solve each exponential equation. Express irrational solutions in exact form and as a decimal rounded to three decimal places. $3 \cdot 4^{x}+4 \cdot 2^{x}+8=0$

Can factorial be extended to negative integers? On the basis of equation (3), what value should (-1)! be given?

(a) The Russell Paradox A logical difficulty arises from the idea, which at first appears natural, of calling any collection of objects a set. Let's say that set B is ordinary if B ff. B. For example, if B is the set of all chairs, then B ff. B, because Bis not a chair. It is only in the case of very unusual collections that we are tempted to say that a set is a member of itself. (The collection of all abstract ideas certainly is an abstract idea.) Let X = { x: x is an ordinary set} . Is X E: X? Is X ff. X? What should we say about the collection of all ordinary sets? (b) In the town of Seville, the (male) barber shaves all the men, and only the men, who do not shave themselves . Let A be the set of all men in the town who do not shave themselves. Who shaves the barber ? (That is, is the barber an element of A? Is he not an element of A?)

Prove: If S is denumerable, then S is equinumerous with a proper subset of itself.

Explain how you can use the number line to compare the rational numbers for Anna and Mike.

Let A and B be sets. Show that a) (A ∩ B) ⊆ A. b) A ⊆ (A ∪ B). c) A − B ⊆ A. d) A ∩ (B − A) = ∅. e) A ∪ (B − A) = A ∪ B.