Download An Introduction to the PL/CV2 Programming Logic by R. L. Constable PDF

By R. L. Constable

ISBN-10: 3540114920

ISBN-13: 9783540114925

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Extra info for An Introduction to the PL/CV2 Programming Logic

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I,3> e I X J} is a partition of A X B. 17. Let G and H be equivalence relations on A. Prove that for each x e A, l>] G f W = [ #1G n [afl^. (Cf. exercise 6) 18. Suppose that G and 77 are equivalence relations on A and that G U # is also an equivalence relation on 4. Prove that for each x e A, [ ] * GW = [a;]Gu 19. Prove that Z [a V := N2/~, with ~ := a+d = b+c, is a commutative ring with unit element and without zero-divisors. 20. c is a . 21. Give an example of a relation, which is transitive and symmetric, but not reflexive.

X Example: Let A = {1,2}, A2 = {3,4} arid A = {7,8,9}. Then A. = {A, x A) X i *1 i 1 A, == (,£/,) X 4 n + i 2 x A3 = ({1,2} x {3,4}) x {7,8,9}. 3 Finally we mention that x is not associative; see Exercise 1. Exercises 1. e. that Ax 2. {B x C) ={A x B) x C does not hold for all sets A,B and C. Prove that {a} x {a} = {{{a}}}. e. that A U (B X C) = (A U B) X (A u C) does not hold for all sets A,B and (7. Let A = {1,2,3,4} , B = {5,6,7} , C = {8,9,10}. Determine A X B (A X C) u B X A t (B X C) , C X f (A u B) X (B X A) , (A U 5) X C, (B U C) .

E. that Ax 2. {B x C) ={A x B) x C does not hold for all sets A,B and C. Prove that {a} x {a} = {{{a}}}. e. that A U (B X C) = (A U B) X (A u C) does not hold for all sets A,B and (7. Let A = {1,2,3,4} , B = {5,6,7} , C = {8,9,10}. Determine A X B (A X C) u B X A t (B X C) , C X f (A u B) X (B X A) , (A U 5) X C, (B U C) . 04 X 5) n (C X £>) = (A X D) n (C X B) AnB=0++(AxC)n{BxC)=0 for all non-empty sets C. Let A ^ 0 and C ± 0. ) ^ - > ^ X C C 5 X Z ) . L e t A, B, C and Z? b e n o n - e m p t y s e t s .

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