By Joseph Breuer, Mathematics, Howard F. Fehr

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54,486,432,000 elements. 4. From the set M = {1,2,3} we shall form all the possible subsets. First there is the null set, M0 = { }. The subsets with only one element are: M11 = {l} M12 = {2}, M13 = {3}. The subsets with two elements are: M21 = {1,2}; M22 = {1,3}; M23 = {2,3}. The improper subset is M3 = {1,2,3}. Thus the set M = {1,2,3} has exactly eight or 23 subsets. Using the formula for the number of combinations of n things used 0,1,2,3,… ,p,…,n at a time,† it is easy to establish the following theorem.

The union of two sets is the set of all elements each of which belongs to at least one of the two sets. The union of two sets is symbolized by M ∪ N (read “the union of M and N”) and the union set contains the elements of M and of N, except that elements contained in both M and N are used only once. 6. The intersection of two sets is the set of elements each of which belongs simultaneously to both sets. The intersection of two sets is symbolized by M ∩ N (read “the intersection of M and N”). 7.

Then we have: of height 2, the fraction = 1; of height 3, the fractions , or 2; of height 4, the fractions , , or 3 (the fraction is not in lowest terms and is eliminated); of height 5, the fractions ,, , or 4, etc. To include the negative rational numbers, we write the corresponding negative number after each positive rational number. The set of all rational numbers can thus be written in the following ordered sequence: In this sequence every rational number has a definite position. The set R is a proper superset of all the various sets of numbers heretofore considered.