Let \(A\) and \(B\) be sets of real numbers. Write the negation of each of the following statements:
For every \(a \in A\), it is true that \(a^2 \in B\) .
For at least one \(a \in A\), it is true that \(a^2 \in B\) .
For every \(a \in A\), it is true that \(a^2 \notin B\) .
For at least one \(a \notin A\), it is true that \(a^2 \in B\) .
Solution
Negating a universal quantifier produces an existential quantifier and vice versa, while negating membership changes \(\in\) to \(\notin\).
The negation is: \[ \text{There exists }a\in A\text{ such that }a^2\notin B. \]
The negation is: \[ \text{For every }a\in A,\quad a^2\notin B. \]
The negation is: \[ \text{There exists }a\in A\text{ such that }a^2\in B. \]
The negation is: \[ \text{For every real number }a\notin A,\quad a^2\notin B. \]