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Phoniex University
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Write the negation of each statement in Practice 2.2. It is important to realize that the order in which quantifiers are used affects the truth value. For example, when talking about real numbers, the statement
Â
is true. That is, given any real number x there is always a real number y that is greater than that x. But the statement
is false, since there is no fixed real number y that is greater than every real number. Thus care must be taken when reading (and writing) quantified statements so that the order of the quantifiers is not inadvertently changed.
Practice 2.2
Rewrite each statement using ∃, ∀, and †, as appropriate.
(a) There exists a positive number x such that x2 = 5.
(b) For every positive number M, there is a positive number N such that N < 1/m.="">
(c) If n ≥ N, then | fn(x) − f(x)| ≤ 3 for all x in A.
(d) No positive number x satisfies the equation f(x) = 5. Having seen several examples of how existential and universal quantifiers are used, let us now consider how quantified statements are negated. Consider the statement Every one in the room is awake. What condition must apply to the people in the room in order for the statement to be false? Must everyone be asleep? No, it is sufficient that at least one person be asleep. On the other hand, in order for the statement Someone in the room is asleep. to be false, it must be the case that everyone is awake. Symbolically, if
Hel-----------lo -----------Sir-----------/Ma-----------dam-----------Tha-----------nk -----------You----------- fo-----------r u-----------sin-----------g o-----------ur -----------web-----------sit-----------e a-----------nd -----------acq-----------uis-----------iti-----------on -----------of -----------my -----------pos-----------ted----------- so-----------lut-----------ion-----------.Pl-----------eas-----------e p-----------ing----------- me----------- on-----------cha-----------t I----------- am----------- on-----------lin-----------e o-----------r i-----------nbo-----------x m-----------e a----------- me-----------ssa-----------ge -----------I w-----------ill----------- be-----------