Compound Boolean Statements

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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) $P \land Q$
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.
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.
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.
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).
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).
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.
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.
9.
In how many of the 4 cases of $P$ and $Q$ is $P \lor \neg Q$ true?
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
11.
Find the truth value of $(P \land Q) \lor \neg R$ for $P = 1$ , $Q = 0$ and $R = 1$ .
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).
13.
In how many of the 4 cases of $P$ and $Q$ is $\neg P \land Q$ true?
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) $R \land \neg P \lor Q$
15.
In how many of the 8 cases of $P$ , $Q$ and $R$ is $(P \land Q) \lor R$ true?
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).
17.
In how many of the 8 cases of $P$ , $Q$ and $R$ is $(P \lor \neg Q) \land R$ true?
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.