The negation of.docx - 4 The negation of Every...

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4The negation ofEvery differentiable function is continuous(think of f D C(f))isNot (every differentiable function is continuous),i.e. ¬³fD C(f)´, and isequivalent toSome differentiable function is not continuous, i.e. f D ¬C(f).orThere exists a non-continuous differentiable function, which is also written f D ¬C(f).5The negation of “all elephants are pink”, i.e. of x E P(x), is “not allelephants are pink”, i.e. ¬(xE P(x)), and an equivalent statement is “thereexists an elephant which is not pink”, i.e. xE ¬P(x).The negation of “there exists a pink elephant”, i.e. of x E P(x), is equivalent to “allelephants are not pink”, i.e. xE ¬P(x).This last statement is often confused in every-day discourse with the statement “notall elephants are pink”, i.e. ¬³xE P(x)´, although it has quite a different meaning, and isequivalent to “there is a non-pink elephant”, i.e. xE ¬P(x). For example, if there were