Compound Boolean Statements

Show your work for each problem.

1.
Let $P$ : "The dog is hungry" and $Q$ : "The dog is tired". Which symbols mean "The dog is hungry and tired"?
1. a) $P \land \neg Q$
2. b) $\neg P \land Q$
3. c) $P \lor Q$
4. d) $P \land Q$
Answer: $P \land Q$
Both parts are claimed to be true and joined with *and*, so the statement is $P \land Q$ .
2.
$P$ is true and $Q$ is false. What is the truth value of $P \lor Q$ ? Write 1 for true and 0 for false.
Answer: 1
$P \lor Q = 1 \lor 0 = 1$ : one true part is enough for OR.
3.
Let $P$ : "It is snowing" and $Q$ : "School is closed". Which sentence matches $\neg P \land Q$ ?
1. a)It is not snowing and school is closed.
2. b)It is snowing and school is not closed.
3. c)It is not true that it is snowing and that school is closed.
4. d)It is not snowing or school is closed.
Answer: It is not snowing and school is closed.
$\neg P$ is "it is not snowing", and $\land$ joins it with $Q$ using *and*. The last choice would be $\neg(P \land Q)$ .
4.
Let $P$ : " $4 + 4 = 8$ " and $Q$ : " $9$ is an even number". What is the truth value of $P \land Q$ ? Write 1 for true and 0 for false.
Answer: 0
$P = 1$ and $Q = 0$ because $9$ is odd. $1 \land 0 = 0$ : AND needs both parts to be true.
5.
Let $P$ : " $5 > 2$ " and $Q$ : " $6$ is an odd number". Find the truth value of $P \land \neg Q$ (1 = true, 0 = false).
Answer: 1
1. What is the value of $P$ ? 1
2. What is the value of $Q$ ? 0
3. What is the value of $\neg Q$ ? 1
4. What is the value of $P \land \neg Q$ ? 1
6.
Let $P$ : "Mia has a cat" and $Q$ : "Mia has a dog". Mia has no pets at all. Find the truth value of "Mia has neither a cat nor a dog" (1 = true, 0 = false).
Answer: 1
1. What is the value of $P$ ? 0
2. Neither ... nor is $\neg P \land \neg Q$ . What is the value of $\neg P$ ? 1
3. What is the value of $\neg Q$ ? 1
4. What is the value of $\neg P \land \neg Q$ ? 1
7.
Which sentence is the negation of "Leo plays the guitar and Leo sings"?
1. a)Leo does not play the guitar and Leo sings.
2. b)It is not true that Leo both plays the guitar and sings.
3. c)Leo neither plays the guitar nor sings.
4. d)Leo plays the guitar but does not sing.
Answer: It is not true that Leo both plays the guitar and sings.
With $G$ : Leo plays the guitar and $S$ : Leo sings, the negation is $\neg(G \land S)$ : it is true in 3 of the 4 cases, whenever at least one of the two is missing. "Neither ... nor" is true in only 1 case, so it is not the negation.
8.
$P$ is false and $Q$ is true. What is the truth value of $\neg(P \lor Q)$ ? Write 1 for true and 0 for false.
Answer: 0
$P \lor Q = 0 \lor 1 = 1$ , and $\neg 1 = 0$ : the statement is false.
9.
In how many of the 4 cases of $P$ and $Q$ is $P \lor \neg Q$ true?
Answer: 3
Cases ( $P$ , $Q$ ): 00 gives $0 \lor 1 = 1$ , 01 gives $0 \lor 0 = 0$ , 10 gives $1$ , 11 gives $1$ . True in 3 cases.
10.
Ana goes to the party and Ben goes to the party. Is the statement "Ana goes or Ben goes" true in logic?
1. a)It cannot be decided
2. b)Yes, because OR is true when at least one part is true
3. c)No, because OR means exactly one of them
4. d)No, because both of them go
Answer: Yes, because OR is true when at least one part is true
In logic OR is inclusive: $1 \lor 1 = 1$ . The statement would be false only if neither Ana nor Ben went.
11.
Find the truth value of $(P \land Q) \lor \neg R$ for $P = 1$ , $Q = 0$ and $R = 1$ .
Answer: 0
1. What is the value of $P \land Q$ ? 0
2. What is the value of $\neg R$ ? 0
3. What is the value of $(P \land Q) \lor \neg R$ ? 0
12.
Let $P$ : " $12$ is divisible by $3$ " and $Q$ : " $12$ is divisible by $5$ ". Find the truth value of " $12$ is divisible by $3$ or by $5$ , but not by both", written $(P \lor Q) \land \neg(P \land Q)$ (1 = true, 0 = false).
Answer: 1
1. What is the value of $P \lor Q$ ? 1
2. What is the value of $P \land Q$ ? 0
3. What is the value of $\neg(P \land Q)$ ? 1
4. What is the value of the whole statement? 1
13.
In how many of the 4 cases of $P$ and $Q$ is $\neg P \land Q$ true?
Answer: 1
1. How many cases are there for $P$ and $Q$ ? 4
2. In how many cases is $\neg P$ true? 2
3. Of those cases, in how many is $Q$ also true? 1
14.
Let $P$ : "It is cold", $Q$ : "It is windy", $R$ : "It is sunny". Which symbols mean "It is sunny, and it is neither cold nor windy"?
1. a) $\neg R \land (P \lor Q)$
2. b) $R \lor \neg(P \land Q)$
3. c) $R \land \neg(P \lor Q)$
4. d) $R \land \neg P \lor Q$
Answer: $R \land \neg(P \lor Q)$
Neither cold nor windy means: it is not true that it is cold or windy, $\neg(P \lor Q)$ , which is true only when $P = 0$ and $Q = 0$ . Joined with sunny by *and*: $R \land \neg(P \lor Q)$ .
15.
In how many of the 8 cases of $P$ , $Q$ and $R$ is $(P \land Q) \lor R$ true?
Answer: 5
When $R = 1$ the statement is always true: 4 cases. When $R = 0$ it needs $P = 1$ and $Q = 1$ : 1 more case. Total $4 + 1 = 5$ .
16.
Let $P$ : " $2^3 = 8$ ", $Q$ : " $15$ is a prime number", $R$ : " $\frac{1}{2} > \frac{1}{3}$ ". Find the truth value of $\neg(P \land R) \lor Q$ (1 = true, 0 = false).
Answer: 0
1. What is the value of $P \land R$ ? 1
2. What is the value of $\neg(P \land R)$ ? 0
3. What is the value of $Q$ ? 0
4. What is the value of $\neg(P \land R) \lor Q$ ? 0
17.
In how many of the 8 cases of $P$ , $Q$ and $R$ is $(P \lor \neg Q) \land R$ true?
Answer: 3
1. How many cases are there for 3 statements? 8
2. In how many of them is $R$ true? 4
3. Of the 4 combinations of $P$ and $Q$ , in how many is $P \lor \neg Q$ true? 3
4. In how many cases is $(P \lor \neg Q) \land R$ true? 3
18.
Let $L$ : "The bus is late" and $W$ : "It is raining". Negate $L \lor W$ and find in how many of the 4 cases the negation is true.
Answer: 1
1. In how many cases is $L \lor W$ true? 3
2. In the case where $L \lor W$ is false, what is the value of $L$ ? (1 = true, 0 = false) 0
3. And what is the value of $W$ in that case? 0
4. In how many cases is $\neg(L \lor W)$ true? 1