Determine whether or not each of the following is a statement or an open sentence. If it is a statement, determine if it is true or false.
If 13 is prime, then 6 is also prime.
If 6 is prime, then 13 is also prime.
\(\displaystyle f(3)=2\)
13 is prime and 6 is prime.
13 is prime or 6 is prime.
The circle’s radius is equal to 1.
2.
Indicate whether the following are true or false.
If today is Saturday, then it is a weekend.
If it is a weekend, then today is Saturday.
If the moon is made of cheese, then every cat in this room is purple.
3.
Indicate whether the following are true or false. Explain your answers.
If \(x\) is even, then \(x\in \{2n : n\in \mathbb{N}\}\text{.}\)
If \(x\) is prime, then \(x= 2k+1\) for some \(k\in \mathbb{Z}\text{.}\)
If \(x\in \{3k : k\in \mathbb{Z}\}\) then \(x\in \{6k : k\in \mathbb{Z}\}\text{.}\)
If \(x\in \{6k : k\in \mathbb{Z}\}\) then \(x\in \{3k : k\in \mathbb{Z}\}\text{.}\)
4.
Indicate whether the following are true or false.
\(3\) is prime and \(3\) is even.
\(3\) is prime or \(3\) is even.
For \(x\in \mathbb{R}\text{,}\)\(x^2 > x\) when \(x>1\text{,}\) and \(18\) is composite.
For \(x\in \mathbb{R}\text{,}\)\(x^2 > x\) when \(x>1\text{,}\) or \(18\) is composite.
5.
Write the following sentences in symbolic logic notation. Make sure to note which statements/open sentences are denoted with which letter.
Example: The sentence, “The car is red and blue but not green” can be written as \((P\land Q)\land (\neg R)\)), where \(P\text{:}\) “The car is red”, \(Q\text{:}\) “The car is blue”, and \(R\text{:}\) “The car is green”. Also, the truth value of this sentence depends on the car, so it is an open sentence, not a statement.
\(8\) is even and \(5\) is prime.
If \(n\) is a multiple of \(4\) and \(6\text{,}\) then it is a multiple of \(24\text{.}\)
If \(n\) is a not a multiple of \(10\text{,}\) then it is a multiple of \(2\) but is not a multiple of \(5\text{.}\)
\(3\leq x\leq 6\text{.}\)
A real number \(x\) is less than \(-2\) or greater than \(2\) if its square is greater than \(4\text{.}\)
If a function \(f\) is differentiable everywhere then whenever \(x\in\mathbb{R}\) is a local maximum of \(f\) we have \(f'(x)=0\text{.}\)
6.
Write the following symbolic statements as English sentences.
Let \(P\) and \(Q\) be statements. Show that the truth table for \(\neg(P\implies Q)\) is the same as the truth table for \(P\land\neg Q\text{.}\)
9.
In each of the following situations, determine whether or not it was raining on the given day, or explain why you cannot determine whether or not it was raining. For each situation we give you two pieces of information that are true; one is an implication and one is a statement.
If it rains, then I bring an umbrella to work. I brought an umbrella to work on Monday.
If it rains, then I bring an umbrella to work. I did not bring an umbrella to work on Tuesday.
Whenever I am late for work, it rains. I was late to work on Wednesday.
Whenever I am late for work, it rains. I was not late to work on Thursday.
10.
There is an old saying: "Red sky at night, sailor’s delight. Red sky at morning, sailors take warning." The phrase tells us that if the sky is red at night, tomorrow’s weather will be good for sailing. However, if the sky is red in the morning, there will be a storm that day, and sailors should be prepared.
Assume that the following statement is true:
\begin{gather*}
\text{If the sky is red and it is morning, then sailors should take warning.}
\end{gather*}
Now assume also that ...
the sky is red. What can we conclude?
the sky is red and it is morning. What can we conclude?
sailors should take warning. What can we conclude?
it is not true that (the sky is red and it is morning). That is, the sky is not red or it is not morning. What can we conclude?
sailors should not take warning. What can we conclude?
11.
Write the contrapositive of the following statements.
If \(n\) is a multiple of \(4\) and \(6\text{,}\) then it is a multiple of \(24\text{.}\)
If \(n\) is not a multiple of \(10\text{,}\) then it is a multiple of \(2\) but is not a multiple of \(5\text{.}\)
A real number \(x\) is less than \(-2\) or greater than \(2\) if its square is greater than \(4\text{.}\)
Let \(m\in\mathbb{N}\text{.}\) Then two true statements are:
\begin{align*}
\text{If }\amp m \text{ is odd, then } m^2 \text{ is odd.}\\
\text{If }\amp m \text{ is even, then } m^2 \text{ is divisible by 4.}
\end{align*}
Construct the contrapositive of each implication to give a total of four different implications. Which combinations can you chain together (so that the conclusion of the first is the hypothesis of the second), and what new implications do these combinations form?
13.
In Chapter 11, we will prove that the following implication is true for \(p=2\text{:}\)
\begin{gather*}
\text{If } p \text{ is prime, then } \sqrt{p} \text{ is irrational.}
\end{gather*}
In fact, this implication is true for any prime number \(p\text{.}\)
Write out the contrapositive, converse, and inverse of this implication. Can you determine whether any of these are true or false statements from the fact that the original implication is true?
14.
Let \(P\text{,}\)\(Q\text{,}\) and \(R\) be statements. Suppose that
“\(P\implies (Q\land R)\)”is false, and
“\(\big((\neg Q)\land R\big) \implies (\neg P)\)” is true.
Which of \(P\text{,}\)\(Q\text{,}\) and \(R\) can you determine are true or false?
15.
Let \(P\text{,}\)\(Q\text{,}\)\(R\text{,}\) and \(S\) be statements. Suppose that
\(S\) is true,
“\(\big( R\lor(\neg P)\big) \implies \big(Q\land (\neg S)\big)\)” is true, and
“\(P \iff \big(Q\lor (\neg S) \big) \)” is true.
Determine the truth values of \(P\text{,}\)\(Q\text{,}\) and \(R\text{.}\)
16.
Let \(P\text{,}\)\(Q\text{,}\)\(R\text{,}\) and \(S\) be statements. Suppose that
“\(((P\lor Q)\implies R) \iff (Q\land S) \)” is true,
“\((P\lor Q)\implies R\)” is false, and
\(S\) is true.
Determine the truth values of \(P\text{,}\)\(Q\text{,}\) and \(R\text{.}\)
