Questions: Let p and q represent the following simple statements: p : The taxes are high. q : I work hard. Write the symbolic statement ¬(p ∧ q) in words. Choose the correct sentence below. A. It is not true that the taxes are high or I work hard. B. It is not true that the taxes are high and I work hard. C. It is not true that the taxes are not high and I do not work hard. D. The taxes are not high and I do not work hard.

Let p and q represent the following simple statements:
p : The taxes are high.
q : I work hard.
Write the symbolic statement ¬(p ∧ q) in words.

Choose the correct sentence below.
A. It is not true that the taxes are high or I work hard.
B. It is not true that the taxes are high and I work hard.
C. It is not true that the taxes are not high and I do not work hard.
D. The taxes are not high and I do not work hard.
Transcript text: Let $p$ and $q$ represent the following simple statements: $p$ : The taxes are high. $q$ : I work hard. Write the symbolic statement $\sim(p \wedge q)$ in words. Choose the correct sentence below. A. It is not true that the taxes are high or I work hard. B. It is not true that the taxes are high and I work hard. C. It is not true that the taxes are not high and I do not work hard. D. The taxes are not high and I do not work hard.
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Solution

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Solution Steps

Step 1: Understand the given symbolic statement

The symbolic statement is \(\sim(p \wedge q)\). Here, \(\sim\) represents negation, and \(\wedge\) represents the logical AND operation. So, \(\sim(p \wedge q)\) means "not (p and q)."

Step 2: Translate the symbolic statement into words

The statement \(\sim(p \wedge q)\) translates to "It is not true that the taxes are high and I work hard."

Step 3: Match the translated statement with the given options
  • A. It is not true that the taxes are high or I work hard.
    This is incorrect because it uses "or" instead of "and."
  • B. It is not true that the taxes are high and I work hard.
    This matches the translated statement.
  • C. It is not true that the taxes are not high and I do not work hard.
    This is incorrect because it introduces additional negations.
  • D. The taxes are not high and I do not work hard.
    This is incorrect because it does not reflect the negation of the entire statement.

Final Answer

The correct sentence is B.
\(\boxed{\text{B}}\)

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