Fundamental Approach to Discrete Mathematics - download pdf or read online

By D. P. Acharjya

ISBN-10: 8122426077

ISBN-13: 9788122426076

The salient positive aspects of this publication comprise: powerful insurance of key issues related to recurrence relation, combinatorics, Boolean algebra, graph conception and fuzzy set thought. Algorithms and examples built-in in the course of the publication to deliver readability to the basic recommendations. each one suggestion and definition is by means of considerate examples. there's straight forward and obtainable presentation to make studying extra fascinating up to attainable with out compromising mathematical rigour. This identify contains thesaurus of all symbols mentioned within the booklet including the bankruptcy the place every one used to be brought. round three hundred whole solved illustrations are there to provide an explanation for the strategies. Over three hundred end-of-chapter routines are integrated to stimulate additional curiosity within the topic.

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I) (A È B) (ii) (A È B) Ç C (iii) (A È B) Ç (A È C) (iv) (A Ç B) È (A Ç C) (v) A – (B È C) (vi) (A Ç B) – C (vii) Bc – (C – A) (viii) Ac Ç Bc (ix) AB (x) Ac (xi) C – B (xii) (A È B) – (C – B) 5. Draw the Venn diagram and indicate the region for the given sets. (a) A È (B Ç C) (b) A Ç (B È C) (c) Ac – B (d) (A È B) – B (e) (Ac È B) Ç (Cc – A) (f) B Ç (C È A)c (g) (B È C) – A (h) (A È B È C)c 6. In a group of 1000 people, there are 800 people who can speak English and 500 people who can speak German.

Solution : Given A = {4, 5}, B = {7, 8} and C = {9, 10} (a) (A ´ B) = {(4, 7), (4, 8), (5, 7), (5,8)} (B ´ C) = {(7, 9), (7, 10), (8, 9), (8, 10)} So, (A ´ B) È (B ´ C) ={(4, 7), (4, 8), (5, 7), (5,8), (7, 9), (7, 10), (8, 9), (8, 10)} (b) (B È C) = {7, 8, 9, 10} So, A ´ (B È C) = {(4, 7), (4, 8), (4, 9), (4,10), (5,7), (5, 8), (5, 9), (5, 10)}. Example 11 If A = {1, 2, 3}, B = {2, 3, 4} and C = {3, 4, 5}, then verify the product laws. Solution : Given A = {1, 2, 3}, B = {2, 3, 4} and C = {3, 4, 5} Therefore (B È C) = {2, 3, 4, 5} and A ´ (B È C) = {(1, 2), (1, 3), (1, 4), (1, 5), (2, 2), (2, 3), (2, 4), (2, 5), (3, 2), (3, 3), (3, 4), (3, 5)} (A ´ B) = {(1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4), (3, 2), (3, 3), (3, 4)} (A ´ C) = {(1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5), (3, 3), (3, 4), (3, 5)} Thus (A ´ B) È (A ´ C) = {(1, 2), (1, 3), (1, 4), (1, 5), (2, 2), (2, 3), (2, 4), (2, 5), (3, 2), (3, 3), (3, 4), (3, 5)} = A ´ (B È C) Similarly the second product law A ´ (B Ç C) = (A ´ B) Ç (A ´ C) can be verified.

A) (A D A) = f (b) (A D B) = (B D A) (c) A Ç (B D C) = (A Ç B) D (A Ç C) (d) (A D B) = (A È B) – (A Ç B) Proof : Proofs of (a) and (b) are immediate consequences of definitions. Here we prove (c) and (d). $ Fundamental Approach to Discrete Mathematics (c) x Î A Ç (B D C) Û x Î A and x Î (B D C) Û x Î A and x Î ((B – C) È (C – B)) Û x Î A and (x Î (B – C) or x Î (C – B)) Û (x Î A and x Î (B – C)) or(x Î A and x Î (C – B)) Û (x Î A and (x Î B and x Ï C)) or (x Î A and (x Î C and x Ï B)) Û ((x Î A and x Î B) and (x Î A and x Ï C)) or ((x Î A and x Î C) and (x Î A and x Ï B)) Û (x Î (A Ç B) and x Ï (A Ç C)) or (x Î (A Ç C) and x Ï (A Ç B)) Û x Î ((A Ç B) – (A Ç C)) or x Î ((A Ç C) – (A Ç B)) Û x Î ((A Ç B) – (A Ç C)) È ((A Ç C) – (A Ç B)) Û x Î (A Ç B) D (A Ç C) So, A Ç (B D C) = (A Ç B) D (A Ç C).

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Fundamental Approach to Discrete Mathematics by D. P. Acharjya


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