In exercises 59-62,let p: Tanisha owns a convertible. q: Joan owns a Volvo. Translate each statement into symbols. Then construct a truth table for each and indicate under what conditions the compound statement is true.#62 Tanisha does not own a convertible or Joan does not own a Volvo.
p: Tanisha owns a convertible
q: Joan owns a Volvo
Tanisha does not own a convertible or Joan does not own a Volvo.
~p v ~q
or
~(p ^ q) (via deMorgan's laws)
To translate the statement "Tanisha does not own a convertible or Joan does not own a Volvo" into symbols, we can use the following notation:
p: Tanisha owns a convertible.
q: Joan owns a Volvo.
The compound statement can be represented as ¬p ∨ ¬q, which means "not ot q."
Next, we can construct a truth table to determine under what conditions the compound statement is true. A truth table lists all possible combinations of truth values for the variables p and q and shows the resulting truth value for the compound statement.
Here is the truth table for ¬p ∨ ¬q:
| p | q | ¬p | ¬q | ¬p ∨ ¬q |
| --- | --- | -- | -- | ------- |
| T | T | F | F | F |
| T | F | F | T | T |
| F | T | T | F | T |
| F | F | T | T | T |
In a truth table, T represents true, and F represents false.
Looking at the truth table, we see that the compound statement ¬p ∨ ¬q is true in three out of the four possible combinations of truth values. It is true when either Tanisha does not own a convertible (¬p) or Joan does not own a Volvo (¬q), or when both of these conditions are true.
Therefore, the compound statement "Tanisha does not own a convertible or Joan does not own a Volvo" is true under the following conditions:
- Tanisha does not own a convertible.
- Joan does not own a Volvo.
- Neither Tanisha nor Joan owns the specified vehicles.