Algebra by Cohn P. M.

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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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 .

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