Lisa says, "If I don’t study, then I will not pass the test." If p: "I study" and q: "I pass the test," which symbolic statement represents Lisa’s statement?
Understand the Problem
The question is asking for the symbolic representation of Lisa's statement, which is conditional in nature. It requires translating a verbal statement into logical notation based on the given definitions of p and q.
Answer
The answer is $$ \neg p \to \neg q $$
Answer for screen readers
The symbolic statement that represents Lisa's statement is:
$$ \neg p \to \neg q $$
Steps to Solve
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Identify the statements
Given that $p$: "I study" and $q$: "I pass the test," we can determine that Lisa's statement involves negations of these conditions. -
Rephrase Lisa’s statement
Lisa's statement "If I don’t study, then I will not pass the test" can be rephrased logically as:
"If not $p$ (I don't study) occurs, then not $q$ (I do not pass the test) occurs." -
Translate into symbolic form
This leads to the symbolic representation:
$$ \neg p \to \neg q $$
This means "If I do not study ($\neg p$), then I do not pass the test ($\neg q$)." -
Match with answer choices
Compare the derived expression with the answer choices provided:- a. $ \neg q \to \neg p$
- b. $ \neg p \to \neg q$
- c. $ p \to q$
- d. $ p \land \neg q$
The correct symbolic statement is option (b): $ \neg p \to \neg q $.
The symbolic statement that represents Lisa's statement is:
$$ \neg p \to \neg q $$
More Information
This type of statement reflects a common logical construction known as "inverse" reasoning. In logical expressions, this means that failing to meet the initial condition directly results in failing the consequent condition.
Tips
- Neglecting negations: Neglecting to recognize that "I don’t study" needs to be represented as $\neg p$.
- Incorrect conditionals: Mixing up the order of implications can lead to incorrect answers, such as confusing $ \neg p \to \neg q$ with $ \neg q \to \neg p$.
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