Circle all the statements below that are equivalent to the negation of the state

Question

Circle all the statements below that are equivalent to the negation of the statement "for all integers x, if x has property P then either x is even or there exists an integer y such that y is greater than x and y does not have property Q. No justification needed. (a) For all integers x, x does not have property P implies either x is even or there exists an integer y such that y is greater than x and y does not have property Q. (b) There is an integer x such that if x has property P then x is odd and for all integers y either y is less than or equal to x or y has property. (c) There is an integer x such that x has property P and x is odd and for all integers y y is less than or equal to x and y has property Q. (d) there is an integer x such that x is not even and for all y y is less than x and y has property Q implies x does not have property P. (e) There is an integer x such that x has property P or x is odd and for all integers y y is less than or equal to x and y has property Q. (f) There is an integer x such that x has property P and x is odd and for all integers y either y is less than or equal to x or y has property.

Explanation / Answer

Both option (a) and (d) is the correct answer.

In option (a) all integersx , x does not have property P implies either x is even or there exists an integer y such that y is greater than x and y does not have property Q.

And in option (d) there is an integer x such that x is not even and for all y , y is less than x and y has property Q implies x does not have property P.

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