Questions: (b) Suppose the following situation occurred. Isabel didn't put money into a savings account each month. When summertime came, Isabel was able to take a vacation. Complete the table to determine the truth of Isabel's mom's statement in this situation. Use T for true and F for false. You may add more columns, bu those added columns will not be graded.

(b) Suppose the following situation occurred. Isabel didn't put money into a savings account each month. When summertime came, Isabel was able to take a vacation. Complete the table to determine the truth of Isabel's mom's statement in this situation. Use T for true and F for false. You may add more columns, bu those added columns will not be graded.
Transcript text: (b) Suppose the following situation occurred. Isabel didn't put money into a savings account each month. When summertime came, Isabel was able to take a vacation. Complete the table to determine the truth of Isabel's mom's statement in this situation. Use T for true and F for false. You may add more columns, bu those added columns will not be graded.
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Solution

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Solution Steps

Step 1: Define the propositions

Let _p_ be the proposition "Isabel put money into a savings account each month." Let _q_ be the proposition "Isabel was able to take a vacation."

The given statement in the problem says that Isabel _didn't_ put money into her savings account and she _was_ able to take a vacation. Thus, _p_ is false and _q_ is true.

Step 2: Evaluate ~p and ~q

~p represents "not p", meaning Isabel did _not_ put money into a savings account each month. Since _p_ is false, ~p is true. ~q represents "not q", meaning Isabel was _not_ able to take a vacation. Since _q_ is true, ~q is false.

Step 3: Evaluate ~p → ~q

~p → ~q is the conditional statement "If not p, then not q". This translates to "If Isabel did not put money into a savings account each month, then she was not able to take a vacation." We know ~p is true and ~q is false. A conditional statement is only false when the hypothesis is true and the conclusion is false. Therefore, ~p → ~q is false.

Final Answer:

| _p_ | _q_ | ~_p_ | ~_q_ | ~_p_ → ~_q_ | |---|---|---|---|---| | F | T | T | F | F |

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