Questions: Use De Morgan's laws to determine whether the two statements are equivalent. ¬(¬y ∨ z) → x, (y ∧ ¬z) → x Choose the correct answer below. The two statements are not equivalent. The two statements are equivalent.

Use De Morgan's laws to determine whether the two statements are equivalent.
¬(¬y ∨ z) → x, (y ∧ ¬z) → x

Choose the correct answer below.
The two statements are not equivalent.
The two statements are equivalent.

Transcript text: Use De Morgan's laws to determine whether the two statements are equivalent. \[ \sim(\sim y \vee z) \rightarrow x,(y \wedge \sim z) \rightarrow x \] Choose the correct answer below. The two statements are not equivalent. The two statements are equivalent.

Solution

Solution Steps

Step 1: Apply De Morgan's Laws to the First Statement

Start with the first statement:
\[ \sim(\sim y \vee z) \rightarrow x \]
Apply De Morgan's laws to \(\sim(\sim y \vee z)\):
\[ \sim(\sim y \vee z) \equiv y \wedge \sim z \]
So, the first statement becomes:
\[ (y \wedge \sim z) \rightarrow x \]

Step 2: Compare the Transformed First Statement with the Second Statement

The second statement is:
\[ (y \wedge \sim z) \rightarrow x \]
After applying De Morgan's laws, the first statement is identical to the second statement.

Step 3: Conclude Equivalence

Since both statements simplify to the same logical expression, they are equivalent.
The correct answer is:
The two statements are equivalent.