By Max Black
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Denoted () C,, is the set of all clements which JCT are in C, for each t E: T. In symbols, () C, tCT = {x; x E: C, for each IE: T}. . .. . . . . . . . . . . . . . . . . . . . . . . . . 4 • 4I If T = {a,b,c,dl, and C. = { l ,2,3} Cb = {2,3,4,S) = {2,3,5,7} Cd = Il ,2,3,S) then n C, = {2,31. As in the case of set union we agree that if c. T = ,er {al, then n tCT C, = c.. Furthermore, if T = {l ,2,3, . . ,nl, we may write the intersection of the sets C, as follows: \Vhen two sets have no elements in common, as in Example (3), it follows that their intersection set is empty.
A il B ) il C 5. 6. 1. 6 C). T H E O R EM S ON S ET S Now that some of the basic terminology of set theory has been defined and two simple theorems have been proved, we will follow a systematic procedure to establish other useful theorems some of which will themselves be machinery for later developments. 3 If A and B are sets, then A U B = BU A. PROOF• Let x E: A VB. 8. 6 • 49 for V. 8. Since x was an arbitrary element of A U B, we conclude that every element of A U Bis an element of BU A. 1. Let y E: BU A.
Then, since xis an arbitrary element of A, conclude that every element of A is an element of B. 1. Next, let y E: B. Again some inference may usually be made, due to the nature of B, which should ultimate:y lead to the conclusion that y E: A . 1. 2. D. WAR NIKG ro not specify x (or y) as a partuular element of A (or B), since we want to draw a conclusion regarding every element of A (or B), and hence must be sure that x (or y) i_s arbitrary. In this and the preceding section we have d iscussed equality of sets, subset, and element of a set.
The Nature of Mathematics by Max Black
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