(Quantifiers.) (a) Without using any words of negation (i.e. without using the w

Question

(Quantifiers.) (a) Without using any words of negation (i.e. without using the words not, no, nor, neither,. . . ), write down a sentence that describes the negation of the following: “If a book on my bookshelf has a page with more than fifty words on it, then the first letter of every word on that page is a vowel.”

(b) A famous quote of Abraham Lincoln goes as follows: “You can fool all the people some of the time, and some of the people all the time, but you cannot fool all the people all the time.” Let P be the set of all people, T be the set of all times, and f(p, t) a function that takes in a person p and time t, and outputs either “fooled by you” or “not fooled by you.” Write this statement using quantifiers and logical/set theory notation.

(c) Let S be a subset of R. Consider the following two propositions: • A: There is a real number M such that for any x S, we have |x| M. • B: For any x S, there is a real number M such that |x| M. Write both of these statements using quantifiers. Is is true that A B? Is is true that B A?

Explanation / Answer

The answers are as follows.

Hope this helps.

(a) If a book on my bookshelf has a page with more htan fifty words on it, then the first letter of every word on that page isn't a vowel.

(b) (p)(t)(f(p, t)) (p)(t)(f(p, t)) ¬(p)(t)(f(p, t))

(c) true that A B (by Archimedian Property).

Then there exists x   R such that |x| <= M for all x R. This means that the set R is bounded from above. Since the set R is non-empty (1 R) the completeness axioms implies that a : = sup S R. Applying proposition ("Let A   R be bounded from below. Then a=inf A if and only if a<=x for any x A and for any epsilon>0 there exists x A such that x-epsilon <a') Epsilon=1 we find that m A satisfies m<= a<=m+1. Since m+1 R we obtain contradiction with the fact htat is an upper bound of R.

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