Select the option that is the negation of ∀x∈Z,(x=0)∨(x^2>0)Option 1 ∃x∈Z:(x!=0)∧¬(x^2>0)Option 2 ∃x∈Z:¬(x=0)∨¬(x^2>0)Option 3 ∀x∈Z,¬((x=0)∨(x^2>0))Option 4 ∀x∈Z,(x!=0)∧(x^2<=0)
Question
Select the option that is the negation of ∀x∈Z,(x=0)∨(x^2>0)Option 1 ∃x∈Z:(x!=0)∧¬(x^2>0)Option 2 ∃x∈Z:¬(x=0)∨¬(x^2>0)Option 3 ∀x∈Z,¬((x=0)∨(x^2>0))Option 4 ∀x∈Z,(x!=0)∧(x^2<=0)
Solution
The correct option is Option 1: ∃x∈Z:(x!=0)∧¬(x^2>0).
Here's why:
The statement ∀x∈Z,(x=0)∨(x^2>0) says that for all integers x, x is either 0 or x^2 is greater than 0.
The negation of this statement would be: there exists an integer x such that x is not 0 and x^2 is not greater than 0 (i.e., x^2 is less than or equal to 0).
This is exactly what Option 1: ∃x∈Z:(x!=0)∧¬(x^2>0) says.
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