Write the statement in symbols using the p and q given below. Then construct a truth table for the symbolic statement and select the best match.
p = At most, 100 guests arrived at the wedding reception.
q = There was a lot of cake left over.
It is not the case that, at most, 100 guests arrived at the wedding reception and there was a lot of cake left over
~p: It is not the case that at most 100 guests arrived at the wedding reception
q: there was a lot of cake left over
~p ^ q
The symbolic statement for the given statement is ¬(p ∧ q).
Truth table for the symbolic statement ¬(p ∧ q):
| p | q | p ∧ q | ¬(p ∧ q) |
|-------|-------|-------|----------|
| False | False | False | True |
| False | True | False | True |
| True | False | False | True |
| True | True | True | False |
The best match for the symbolic statement is "It is not the case that, at most, 100 guests arrived at the wedding reception and there was a lot of cake left over."
To represent the given statement in symbols, we can use the following notation:
p: At most 100 guests arrived at the wedding reception.
q: There was a lot of cake left over.
The statement "It is not the case that, at most, 100 guests arrived at the wedding reception and there was a lot of cake left over" can be symbolically represented as ¬(p ∧ q).
To construct a truth table for this symbolic statement, we need to consider all possible combinations of truth values for p and q, and evaluate the truth value of the statement for each combination. Let's build the truth table:
```
╔════════╦═════════╦═════════════════════╗
║ p ║ q ║ ¬(p ∧ q) ║
╠════════╬═════════╬═════════════════════╣
║ T ║ T ║ F ║
║ T ║ F ║ T ║
║ F ║ T ║ T ║
║ F ║ F ║ T ║
╚════════╩═════════╩═════════════════════╝
```
In this truth table, the column ¬(p ∧ q) represents the negation of the conjunction (p ∧ q). True is represented as T, and False is represented as F.
Now, based on the truth table, we can select the best match for the statement. Looking at the truth values, we see that the statement "It is not the case that, at most, 100 guests arrived at the wedding reception and there was a lot of cake left over" is true in three out of the four possible cases. Therefore, the best match would be "Sometimes true."