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we want to simplify the given statements with quantifiers first we have the negation of for every x there exists a y such that x is greater than or equal to y and y doesnamp;#39;t equal zero to begin simplifying we pass the negation across the quantifiers which changes the quantifier types passing the negation across the quantifiers for every changes to there exists and there exist changes to for every this gives us the logically equivalent statement there exists an x for every y such that the negation of x is greater than or equal to y and y doesnamp;#39;t equal zero and now weamp;#39;ll negate this conjunction using de morganamp;#39;s law looking at the notes below the negation of p and q is logically equivalent to not p or not q applying de morganamp;#39;s law gives us for every x there exists a y such that the negation of x is greater than or equal to y or the negation of y doesnamp;#39;t equal zero the last step is to negate x is greater than or equal to y and negate y doesn
