To determine which expression is equivalent to \( !(x < y || z \neq 9) \), we can apply De Morgan's laws. According to De Morgan's laws, the negation of a disjunction (OR statement) can be rewritten as a conjunction (AND statement) of the negations.
Given: \[ !(x < y || z \neq 9) \]
We can apply De Morgan's law:
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Negate both conditions inside the parentheses: \[ !(x < y) \text{ and } !(z \neq 9) \]
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Negating the first condition \( !(x < y) \) gives us: \[ x \geq y \] (because if \( x < y \) is false, then \( x \) must be greater than or equal to \( y \))
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Negating the second condition \( !(z \neq 9) \) gives us: \[ z == 9 \] (because if \( z \neq 9 \) is false, then \( z \) must equal 9)
Putting it all together: \[ !(x < y || z \neq 9) \] is equivalent to: \[ x \geq y \text{ and } z == 9 \]
So the expression you are looking for is: \( x \geq y && z == 9 \).