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# sols1 - 1 AUGUST 30TH BASIC NOTATION QUANTIFIERS 1 1 August...

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Unformatted text preview: 1. AUGUST 30TH: BASIC NOTATION, QUANTIFIERS 1 1. August 30th: Basic notation, quantifiers 1.1. Homework. 1.1 . Assume that P ( x ) is some statement whose truth depends on x . Which of the following is ‘grammatically correct’? • P ( x ) , ∀ • ∀ x, P ( x ) • ∀ x ∈ P ( x ) Answer: To fix ideas, assume that P ( x ) is the statement ‘ x is a rational number’. The first sentence would read “ x is a rational number, for all” and it sounds pleasantly zen, but I don’t know what it could possibly mean. The dangling ‘for all’, by itself, means nothing. The third one says “For all x in x is a rational number”, and again I don’t know what this means: ‘ x is a rational number’ is a statement, not a set. The second one, “For all x , x is a rational number”, makes sense. It may be true or false, depending on the context (we don’t know what x denotes here), but its meaning is clear. 1.2 . Let S be an indexed set of real numbers: S = { s n | n ∈ N } ....
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sols1 - 1 AUGUST 30TH BASIC NOTATION QUANTIFIERS 1 1 August...

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