Discrete Mathematics with Graph Theory (3rd Edition) 43

Discrete Mathematics with Graph Theory (3rd Edition) 43 -...

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Section 2.4 (b) [BB] I = {a I a'" I} = {a I I E Q} = {a I a E Q} = Q" {O}. (c) [BB] 1/ = ~ = 2 E Q, so y'3 '" v'l2 and hence y'3 = v'l2. 41 6. Reflexive: For any a E N, a'" a since a 2 +a = a(a+ 1) is even, as the product of consecutive natural numbers. Symmetric: If a '" b, then a 2 + b is even. It follows that either a and b are both even or both are odd. If they are both even, b 2 + a is the sum of even numbers, hence, even. If they are both odd, b 2 + a is the sum of odd numbers and, hence, again, even. In both cases b 2 + a is even, so b '" a. Transitive: If a '" band b '" c, then a 2 + b and b 2 + c are even, so (a 2 + b) + (b 2 + c) is even; in other words, (a 2 + c) + (b 2 + b) is even. Since b 2 + b is even, a 2 + c is even too; therefore, a '" c. The quotient set is the set of equivalence classes. Now 2. {evens a= {x I x +alseven} = odds So A/",= {2Z, 2Z + I}. if a is even if a is odd 7. (a) [BB] Reflexive: For any a E R, a '" a because a - a = 0 E Z. Symmetric:
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