1.
9 is even and 9 is prime.
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Worksheet Statements and the Search for Truth
Consider three statements: \(P\text{:}\) 9 is even; \(Q\text{:}\) 9 is prime; \(R\text{:}\) 9 is square.
For each molecular statement below, decide whether you think it is true or false based on your understanding of of the number 9. Then write down the statement using the variables \(P\text{,}\) \(Q\text{,}\) and \(R\) with appropriate logical connectives. Finally, verify your original answer by checking the truth table definition of the connectives.
As a reminder, the truth tables for the logical connectives are as follows:
| \(P\) | \(Q\) | \(P\wedge Q\) |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
| \(P\) | \(Q\) | \(P\vee Q\) |
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
| \(P\) | \(Q\) | \(P\imp Q\) |
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
| \(P\) | \(Q\) | \(P \iff Q\) |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
| \(P\) | \(\neg P\) |
| T | F |
| F | T |
2.
9 is even or 9 is square.
3.
9 is not even or 9 is square.
4.
If 9 is even, then 9 is not prime.
5.
If 9 is prime, then 9 is square.
6.
9 is not even if and only if 9 is square.
7.
9 is even and 9 is prime, or 9 is square.
Truth and Quantifiers.
Notation: \(\forall x\) is read “for all x” and \(\exists x\) is read “there exists an x”.
Consider the following sets:
\begin{equation*}
A = \{2, 4, 6, 8\} \qquad B = \{1, 3, 5, 7, 9\} \qquad C = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}
\end{equation*}
Recall that \(6 \in A\) is the statement that \(6\) is an element in the set \(A\) (which is true statement in this case).
For each statement below, decide whether you think it is true or false. Why do you say that? Then write down the statement either in words or using symbols (whichever is not already given).
8.
\(\forall x (x \in A \lor x \in B)\)
9.
\(\exists x (x \in A \land x \in B)\)
10.
\(\exists x (x \in A) \land \exists x (x \in B)\)
11.
\(\exists x (x \in A \lor x \in B)\)
12.
\(\forall x ( x \in A \imp x \in C )\)
13.
Every element in \(C\) is either in \(A\) or in \(B\text{.}\)
14.
There exists an element in \(C\) that is not in \(A\) and not in \(B\text{.}\)
