By Cohn P. M.

Stories very important heritage in algebra and introduces extra complicated subject matters, emphasizing linear algebra and the houses of teams and jewelry. contains extra labored difficulties and a whole set of solutions to the routines. additionally positive factors extended proofs and extra in-depth remedies of affine areas, linear programming, duality, Jordan basic shape and staff concept.

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**Example text**

3. (A U B) UC = A U (BU C). = A n (B il C). A U (8 n C) = (A U B) n (A U A ll (B - A) = f2f. AU (B - A) = AU B. 4. (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.

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.

1. 1. D. by the Axiom of Extent, A = B. 2 is extremely valuable in proving two sets equal. 2 to which the reader may wish to refer from time to time. 2 TO PROVE A = B. PROOF let x E: A. From this some inference may usually be made, due to the nature of set A, which should ultimately lead to the conclusion that x E: B. 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 .