Communications. A cable television company has $8,000$ subscribers in a suburban community. The company offers two premium channels: HBO and Showtime. If $2,450$ subscribers receive HBO, $1,940$ receive Showtime, and $5,180$ do not receive any premium channel, how many subscribers receive both HBO and Showtime?

An entertainment guide recommends 6 restaurants and 3 plays that appeal to a couple. (A) If the couple goes to dinner or a play, but not both, how many selections are possible? (B) If the couple goes to dinner and then to a play, how many combined selections are possible?

A player is dealt two cards from a 52-card deck. If the first card is black, the player returns it to the deck before drawing the second card. If the first card is red, the player sets it aside and then draws the second card. If the second card drawn is black, what is the probability that the first card drawn was black?

A pointer is spun on a circular spinner. The probabilities of the pointer landing on the integers from 1 to 5 are given in the table below.

$$\begin{matrix} \text{}{e_{i}} & \text{1} & \text{2} & \text{3} & \text{4} & \text{5}\\ \text{}{p_{i}} & \text{.2} & \text.{1} & \text{.3} & \text{.3} & \text{.1}\\ \end{matrix}$$

(A) What is the probability of the pointer landing on an odd number? (B) What is the probability of the pointer landing on a number less than 4 given that it landed on an odd number?

Construct a truth table to verify the implication or equivalence.

$$p\land q\implies p$$

Determine whether the given set is disjoint. Consider the set $N$ of positive integers to be the universal set, and let

$$\begin{aligned} H&=\{n\in N|\ n>100\}\\ T&=\{n\in N|\ n< 1,000\}\\ E&=\{n\in N|\ n \text{ is even}\}\\ P&=\{n \in N|\ n \text{ is prime}\} \end{aligned}$$

$H^{\prime}$ and $T^{\prime}$

Determine whether the given set is finite or infinite. Consider the set $N$ of positive integers to be the universal set, and let

$$\begin{aligned} H&=\{n\in N|\ n>100\}\\ T&=\{n\in N|\ n< 1,000\}\\ E&=\{n\in N|\ n \text{ is even}\}\\ P&=\{n \in N|\ n \text{ is prime}\} \end{aligned}$$

$$(E \cup P)^\prime$$

Discuss the validity of each statement. If the statement is true, explain why. If not, give a counterexample.

If $n$ and $r$ are positive integers and $1<r <n$, then $_nC_r < _nC_{r+1}$

Construct a truth table to verify each implication.

$$(p \land \neg p)\implies q$$

Market research. A survey of $800$ small businesses indicates that $250$ own a video conferencing system, $420$ own projection equipment, and $180$ own a video conferencing system and projection equipment.

(A) How many businesses in the survey own either a video conferencing system or projection equipment?

(B) How many own neither a video conferencing system nor projection equipment?

(C) How many own projection equipment and do not own a video conferencing system?

Construct a truth table to verify each implication.

$$\neg p \rightarrow (q\land \neg q)\implies p$$

Determine whether the given set is finite or infinite. Consider the set $N$ of positive integers to be the universal set, and let

$$\begin{aligned} H&=\{n\in N|\ n>100\}\\ T&=\{n\in N|\ n< 1,000\}\\ E&=\{n\in N|\ n \text{ is even}\}\\ P&=\{n \in N|\ n \text{ is prime}\} \end{aligned}$$

$$H^\prime \cup T$$

Market research. A survey of $1,200$ people indicates that $850$ own HDTV’s, $740$ own DVD players, and $580$ own HDTV’s and DVD players.

(A) How many people in the survey own either an HDTV or a DVD player?

(B) How many own neither an HDTV nor a DVD player?

(C) How many own an HDTV and do not own a DVD player?

Discuss the validity of each statement. If the statement is true, explain why. If not, give a counterexample.

If $n$ and $r$ are positive integers and $1<r <n$, then $_nP_r < _nP_{r+1}$