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By Peter W. And Johnson, Robert Zehna

A truly brief advent to Set conception for undergraduate scholars.

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In the illustrations preceding the definitions it should be observed that set D is not only a subset of B but also equals B. On the other hand, sets A and C, though subsets, are not equal to set B. ·we will wish to make a distinction between these two kinds of subsets and therefore make the following definition. 3 Set A is said to be a proper S11bset of set B, denoted A C B, if and on ly if A s; Band there exists an element of B which is not in A. We write, (AC B}~ (AC Band 3 xE: B 3 x(/. A}. ~ Set A is said to be an improp~ subset of set B if and only if A = B.

It is clear from elementary algebra that there are two and only two elements in set C, namely 1 and -2. " Hence C may be written, C = {x; x = 1 or x = - 21. In practice, when the number of elements in a set is small and when they have convenient names, we will simply list these names within braces and use the resulting notation as another name for the set. Thus, we write, C = {1,- 2}. EXAMPLE 13) D = {0,2,4,6,8}. This indicates that D is the set of integers whose numerals appear within the braces.

D. 6 If A, B, and C are sets, then (A n B) n C = A n (B n C) We will leave the proof of this theorem to the reader as an exercise at the end of this section. 5. If we think of sets as analogous to real numbers, set union as analogous to addition of real numbers, and set intersection as analogous to multiplication of numbers, then we notice that the theorems just proved are analogous to the commutative and associative laws of addition and multiplication of real numbers. , we could replace each occurrence of + by · and each occurrence of · by + and still have true statements.

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Elements of Set Theory by Peter W. And Johnson, Robert Zehna

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