Negation: The power goes off and the food does not spoil.
Converse: If the food spoils, then the power went off.
Contrapositive: If the food does not spoil, then the power did not go off.
Negation: The door is closed and the light is on.
Converse: If the light is off then the door is closed.
Contrapositive: If the light is on then the door is open.
Negation: \(\exists x (x \lt 1 \wedge x^2 \ge 1)\)
Converse: \(\forall x( x^2 \lt 1 \imp x \lt 1)\)
Contrapositive: \(\forall x (x^2 \ge 1 \imp x \ge 1)\text{.}\)
Negation: There is a natural number \(n\) which is prime but not solitary.
Converse: For all natural numbers \(n\text{,}\) if \(n\) is solitary, then \(n\) is prime.
Contrapositive: For all natural numbers \(n\text{,}\) if \(n\) is not solitary then \(n\) is not prime.
Negation: There is a function which is differentiable and not continuous.
Converse: For all functions \(f\text{,}\) if \(f\) is continuous then \(f\) is differentiable.
Contrapositive: For all functions \(f\text{,}\) if \(f\) is not continuous then \(f\) is not differentiable.
Negation: There are integers \(a\) and \(b\) for which \(a\cdot b\) is even but \(a\) or \(b\) is odd.
Converse: For all integers \(a\) and \(b\text{,}\) if \(a\) and \(b\) are even then \(ab\) is even.
Contrapositive: For all integers \(a\) and \(b\text{,}\) if \(a\) or \(b\) is odd, then \(ab\) is odd.
Negation: There are integers \(x\) and \(y\) such that for every integer \(n\text{,}\) \(x \gt 0\) and \(nx \le y\text{.}\)
Converse: For every integer \(x\) and every integer \(y\) there is an integer \(n\) such that if \(nx > y\) then \(x > 0\text{.}\)
Contrapositive: For every integer \(x\) and every integer \(y\) there is an integer \(n\) such that if \(nx \le y\) then \(x \le 0\text{.}\)
Negation: There are real numbers \(x\) and \(y\) such that \(xy = 0\) but \(x \ne 0\) and \(y \ne 0\text{.}\)
Converse: For all real numbers \(x\) and \(y\text{,}\) if \(x = 0\) or \(y = 0\) then \(xy = 0\)
Contrapositive: For all real numbers \(x\) and \(y\text{,}\) if \(x \ne 0\) and \(y \ne 0\) then \(xy \ne 0\text{.}\)
Negation: There is at least one student in Math 228 who does not understand implications but will still pass the exam.
Converse: For every student in Math 228, if they fail the exam, then they did not understand implications.
Contrapositive: For every student in Math 228, if they pass the exam, then they understood implications.