Pure Mathematics

Elementary set theory by K.T. Leung

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By K.T. Leung

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Who has not at some time posed questions of the following kind? Are there more whole numbers than there are even numbers? Does an unbounded straight line contain more points than a line segment? Does a plane contain fewer points than space? Are the rational points densely situated on the number scale? In particular, what do ∞ + 1, and ∞ ·3, and ∞2 denote? People refrained from discussing these questions publicly since such inquiries seemed naive or stupid and, above all, because they appeared to have no answer.

Determine the cardinal number of the set of all powers mn where m and n are natural numbers. ) 6. Prove theorem (b) of the previous Section 12. Hint: for a denumerable number of denumerable sets use the diagonal process. VII. Further Non-denumerable Sets 1. Up to this point we have learned of two levels of infinity; of two different kinds of cardinality of infinite sets, and of two transfinite cardinal numbers, a and c. We now seek others. Consider the set, F, of all real functions, defined by y = f(x) in the interval, 0 < x < 1.

If however, you attempt to establish the equality of the cardinal numbers by counting, your attempt would fail, for the sets N and G are infinite sets. The number of elements that each set contains is infinitely large. }, forms an unbounded set (one without end or infinite) of definite and distinct elements, the number of which exceeds every finite cardinal number. Indeed, in this case we can no longer speak of the “number” of elements in the usual sense. ” However, the equivalence concept gives us the means to resolve this question.

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