By Joseph Breuer, Mathematics, Howard F. Fehr

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**Introduction to the Theory of Sets**

Set conception permeates a lot of latest mathematical inspiration. this article for undergraduates bargains a typical creation, constructing the topic via observations of the actual international. Its innovative improvement leads from finite units to cardinal numbers, countless cardinals, and ordinals. workouts look through the textual content, with solutions on the finish.

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The set of all positive odd numbers: U = {1,3,5,7,9,11,…}. The set of all prime numbers: P = {2,3,5,7,11,…}. The set of squares of all integers: Q = {1,4,9,16,25,…}. The set of all integers: Z = {0,1,–1,2,–2,…}. The set of all rational numbers: The set of all algebraic numbers: The set of all lattice points on a straight line: Gg. The set of all lattice points on a plane: Ge. For all of these sets we have the inclusion relations: All of these sets are equivalent: All of these sets have the same cardinal number; they are all denumerable: (b) Sets with the cardinal number of the continuum: The set of all real numbers in the interval 0 < r < 1:.

It is therefore not surprising to find that the set of all differentiable functions also has the cardinal number of the continuum, but on the other hand, the set of all integrable functions has the cardinal number f of the set of all* functions, F. The differentiable functions are again the “exceptional cases” of the integrable functions. ” In this sense (sin x)/x is an integrable function. 4. Are there cardinal numbers beyond a, c, and f? The answer is that there are infinitely many transfinite cardinal numbers.

Figure 14. Figure 15. Figure 16. The set of all positive even integers: G = {2,4,6,8,10,…}. The set of all positive odd numbers: U = {1,3,5,7,9,11,…}. The set of all prime numbers: P = {2,3,5,7,11,…}. The set of squares of all integers: Q = {1,4,9,16,25,…}. The set of all integers: Z = {0,1,–1,2,–2,…}. The set of all rational numbers: The set of all algebraic numbers: The set of all lattice points on a straight line: Gg. The set of all lattice points on a plane: Ge. For all of these sets we have the inclusion relations: All of these sets are equivalent: All of these sets have the same cardinal number; they are all denumerable: (b) Sets with the cardinal number of the continuum: The set of all real numbers in the interval 0 < r < 1:.