The NOT Operation

Show your work for each problem.

1.
If $P$ is true, what is $\neg P$ ?
1. a)False
2. b)True
3. c)Both true and false
4. d)It depends on Q
Answer: False
NOT turns true into false, so $\neg P$ is false. NOT has only one input, so Q plays no role.
2.
With 1 for true and 0 for false, what is NOT 1? Type 1 or 0.
Answer: 0
NOT 1 = 0. You can also use the trick $\neg P = 1 - P$ : $1 - 1 = 0$ .
3.
Which sentence is the negation of "Tom likes football"?
1. a)Tom likes football a lot
2. b)Tom hates football
3. c)Tom likes basketball
4. d)Tom does not like football
Answer: Tom does not like football
"Tom does not like football" is the negation. "Tom hates football" says much more: Tom could simply not care about football, and then he neither likes nor hates it.
4.
What is $\neg(\neg 1)$ ? Type 1 or 0 (1 = true, 0 = false).
Answer: 1
$\neg 1 = 0$ and then $\neg 0 = 1$ . Two NOTs cancel out, so you get back the starting value 1.
5.
Let P = " $4 + 4 = 8$ ". Find the value of P and of $\neg P$ , using 1 for true and 0 for false.
Answer: 0
1. Is P true (1) or false (0)? 1
2. What is $\neg P$ ? 0
6.
Use the trick $\neg P = 1 - P$ to compute NOT 0 and NOT 1.
Answer: 1, 0
1. Compute NOT 0 = $1 - 0$ 1
2. Compute NOT 1 = $1 - 1$ 0
7.
What is the negation of "All birds can fly"?
1. a)Some birds can fly
2. b)All birds are unable to fly
3. c)Not all birds can fly
4. d)No bird can fly
Answer: Not all birds can fly
One bird that cannot fly (a penguin) already makes the claim false, so the negation is "Not all birds can fly". "No bird can fly" is false too, because most birds can fly.
8.
P = 0 and Q = 0. What is $\neg P \vee Q$ ? Type 1 or 0 (1 = true, 0 = false).
Answer: 1
$\neg P = \neg 0 = 1$ , and then $1 \vee 0 = 1$ , because OR needs only one true input.
9.
The negation of " $x > 5$ " is:
1. a) $x \geq 5$
2. b) $x \leq 5$
3. c) $x \neq 5$
4. d) $x < 5$
Answer: $x \leq 5$
A number is either greater than 5 or not. "Not greater than 5" means smaller than 5 or equal to 5, so the negation is $x \leq 5$ . The choice $x < 5$ forgets the number 5.
10.
P = 1 and Q = 0. Evaluate $\neg(P \land Q)$ , then compare it with $\neg P \land Q$ (1 = true, 0 = false).
Answer: 1
1. What is $P \land Q$ ? 0
2. What is $\neg(P \land Q)$ ? 1
3. Now compute $\neg P \land Q$ 0
11.
P = "10 is divisible by 3" and Q = "10 is even". Evaluate $\neg P \land Q$ (1 = true, 0 = false).
Answer: 1
1. What is P? 0
2. What is $\neg P$ ? 1
3. What is Q? 1
4. What is $\neg P \land Q$ ? 1
12.
What is $\neg\neg\neg 0$ (three NOTs in a row)? Type 1 or 0 (1 = true, 0 = false).
Answer: 1
$\neg 0 = 1$ , $\neg 1 = 0$ , $\neg 0 = 1$ . An odd number of NOTs flips the value, an even number keeps it.
13.
Let S = "Mia speaks French", and S is true. What is the value of the sentence "It is not true that Mia does not speak French"? Use 1 for true and 0 for false.
Answer: 1
1. What is the value of S? 1
2. "Mia does not speak French" is $\neg S$ . What is its value? 0
3. The whole sentence is $\neg(\neg S)$ . What is its value? 1
14.
Which expression always has the same value as $\neg(P \vee Q)$ ?
1. a) $\neg P \land \neg Q$
2. b) $\neg P \vee \neg Q$
3. c) $\neg P \vee Q$
4. d) $P \land Q$
Answer: $\neg P \land \neg Q$
By De Morgan's law, $\neg(P \vee Q) = \neg P \land \neg Q$ : "neither P nor Q". Check P = 1, Q = 0: $\neg(1 \vee 0) = 0$ and $\neg 1 \land \neg 0 = 0 \land 1 = 0$ , while $\neg P \vee \neg Q = 0 \vee 1 = 1$ is different.
15.
P = 1, Q = 0 and R = 1. What is $\neg(P \land \neg Q) \vee \neg R$ ? Type 1 or 0 (1 = true, 0 = false).
Answer: 0
$\neg Q = 1$ , so $P \land \neg Q = 1 \land 1 = 1$ and $\neg(P \land \neg Q) = 0$ . Also $\neg R = 0$ . Then $0 \vee 0 = 0$ .
16.
P = 1 and Q = 0. Compute $\neg(P \land Q)$ and $\neg P \vee \neg Q$ and check that they are equal (De Morgan).
Answer: 1
1. What is $P \land Q$ ? 0
2. What is $\neg(P \land Q)$ ? 1
3. What is $\neg P$ ? 0
4. What is $\neg Q$ ? 1
5. What is $\neg P \vee \neg Q$ ? 1
17.
With 1 and 0, AND is multiplication and NOT P is $1 - P$ . Compute NOT (1 AND 0) as $1 - 1 \times 0$ , then check the result with $\neg 1 \vee \neg 0$ .
Answer: 1
1. What is 1 AND 0 = $1 \times 0$ ? 0
2. What is $1 - 0$ ? 1
3. Check: what is $\neg 1 \vee \neg 0$ ? 1
18.
A garden sprinkler runs when $\neg R \land \neg W$ is true, where R = "it is raining" and W = "the soil is already wet". By De Morgan this is the same as $\neg(R \vee W)$ . Today R = 0 and W = 1. Does the sprinkler run? Use 1 for yes (true) and 0 for no (false).
Answer: 0
1. What is $\neg R$ ? 1
2. What is $\neg W$ ? 0
3. What is $\neg R \land \neg W$ ? 0
4. Now the other form. What is $R \vee W$ ? 1
5. What is $\neg(R \vee W)$ ? 0