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By Goodman F.M.
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C) Every element a of Z has an additive inverse a, satisfying a C . a/ D 0. We write a b for a C . b/. (d) Multiplication on Z is commutative and associative. (e) 1 is an identity element for multiplication; that is, for all a 2 Z, 1a D a. b C c/ D ab C ac: (g) (h) N is closed under addition and multiplication. That is, the sum and product of positive integers is positive. The product of non-zero integers is non-zero. We write a > b if a b > 0 and a b if a b 0. Then > is a total order on the integers.
13. Suppose X is the union of disjoint sets X1 and X2 , X D X1 [ X2 and X1 \ X2 D ;. X /. Xi / for i D 1; 2, and (noticing the abuse of notation) also write i for the permutation of X that is i on Xi and the identity on X n Xi . Show that D 1 2 D 2 1 . 14. X/, where X is a finite set. Let x0 be some element of X that is not fixed by . x1 /, and so forth. xk / D x0 . xk / D xkC1 2 fx0 ; x1 ; : : : ; xk g. xk / D x0 . xk / D xl for some l, 1 Ä l Ä k leads to a contradiction. Show that X1 D fx0 ; x1 ; : : : ; xk g and X2 D X n X1 are both invariant under .
5. jmj; jnj/. 6. m; n/ is the largest natural number dividing m and n. 7. m; n/ \ N. 8. m; n/. 9. Show that if p is a prime number and a is any nonzero integer, then either p divides a or p and a are relatively prime. 10. Suppose that a and b are relatively prime integers and that x is an integer. Show that if a divides the product bx, then a divides x. Hint: Use the existence of s; t such that sa C t b D 1. 11. Suppose that a and b are relatively prime integers and that x is an integer. Show that if a divides x and b divides x, then ab divides x.
Algebra. Abstract and concrete by Goodman F.M.