Question

conditional \\(p \
ightarrow q\\)
inverse \\(\sim p \
ightarrow \sim q\\)
converse \\(q \
ightarrow p\\)
contrapositive \\(\sim q \
ightarrow \sim p\\)

multiple choice
question 12
read the conditional statement below:
\if today is monday, then tomorrow is tuesday.\
what is the inverse of this statement?
a if tomorrow is tuesday, then today is monday.
b if tomorrow is not tuesday, then today is not monday.
c if today is not monday, then tomorrow is not tuesday.

Explanation:

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<reasoning>

Identify the conditional statement components

The given conditional statement is: "If today is Monday, then tomorrow is Tuesday."
Let \(p\) represent the hypothesis: "today is Monday."
Let \(q\) represent the conclusion: "tomorrow is Tuesday."
The conditional statement is represented symbolically as \(p
ightarrow q\).

Apply the definition of the inverse

The top of the image provides the definition of the inverse:
Inverse: \(\sim p
ightarrow \sim q\)
This means we negate both the hypothesis (\(p\)) and the conclusion (\(q\)) without swapping their order.

Formulate the negated components

Negating the hypothesis \(p\) ("today is Monday") gives:
\(\sim p\): "today is not Monday."

Negating the conclusion \(q\) ("tomorrow is Tuesday") gives:
\(\sim q\): "tomorrow is not Tuesday."

Construct the inverse statement

Combining these negated components into an "If..., then..." structure yields:
"If today is not Monday, then tomorrow is not Tuesday."

Match with the given options

Let's evaluate the options:

• Option A: "If tomorrow is Tuesday, then today is Monday." (This is the converse, \(q

ightarrow p\))

• Option B: "If tomorrow is not Tuesday, then today is not Monday." (This is the contrapositive, \(\sim q

ightarrow \sim p\))

• Option C: "If today is not Monday, then tomorrow is not Tuesday." (This is the inverse, \(\sim p

ightarrow \sim q\))

Therefore, Option C is the correct statement.
</reasoning>

<answer>
<mcq-option>A If tomorrow is Tuesday, then today is Monday.</mcq-option>
<mcq-option>B If tomorrow is not Tuesday, then today is not Monday.</mcq-option>
<mcq-correct>C If today is not Monday, then tomorrow is not Tuesday.</mcq-correct>
</answer>

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Answer:

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<reasoning>

Identify the conditional statement components

The given conditional statement is: "If today is Monday, then tomorrow is Tuesday."
Let \(p\) represent the hypothesis: "today is Monday."
Let \(q\) represent the conclusion: "tomorrow is Tuesday."
The conditional statement is represented symbolically as \(p
ightarrow q\).

Apply the definition of the inverse

The top of the image provides the definition of the inverse:
Inverse: \(\sim p
ightarrow \sim q\)
This means we negate both the hypothesis (\(p\)) and the conclusion (\(q\)) without swapping their order.

Formulate the negated components

Negating the hypothesis \(p\) ("today is Monday") gives:
\(\sim p\): "today is not Monday."

Negating the conclusion \(q\) ("tomorrow is Tuesday") gives:
\(\sim q\): "tomorrow is not Tuesday."

Construct the inverse statement

Combining these negated components into an "If..., then..." structure yields:
"If today is not Monday, then tomorrow is not Tuesday."

Match with the given options

Let's evaluate the options:

• Option A: "If tomorrow is Tuesday, then today is Monday." (This is the converse, \(q

ightarrow p\))

• Option B: "If tomorrow is not Tuesday, then today is not Monday." (This is the contrapositive, \(\sim q

ightarrow \sim p\))

• Option C: "If today is not Monday, then tomorrow is not Tuesday." (This is the inverse, \(\sim p

ightarrow \sim q\))

Therefore, Option C is the correct statement.
</reasoning>

<answer>
<mcq-option>A If tomorrow is Tuesday, then today is Monday.</mcq-option>
<mcq-option>B If tomorrow is not Tuesday, then today is not Monday.</mcq-option>
<mcq-correct>C If today is not Monday, then tomorrow is not Tuesday.</mcq-correct>
</answer>

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