Let R be the relation defined on Z (integers): a R b iff a + b is even. Then the distinct equivalence classes are: Group of answer choices [1] = multiples of 3 [2] = multiples of 4 [0] = even integers and [1] = the odd integers all the integers None of the above
Let R be the relation defined on Z (integers): a R b iff a + b is even. R is an equivalence relation since R is: Group of answer choices Reflexive, Symmetric and Transitive Symmetric and Reflexive or Transitive Reflexive or Transitive Symmetric and Transitive None of the above
Please explain every step. 29) Let * be the binary operation defined on rational numbers by a*b = a+b-ab. How many integers have integer *-inverses? a) 0 b) 1 c) 2. d) 3 e) 4 None
Let R be the relation defined on Z (integers): a R b iff a + b is even. Suppose that 'even' is replaced by 'odd' . Which of the properties reflexive, symmetric and transitive does R possess? Group of answer choices Reflexive, Symmetric and Transitive Symmetric Symmetric and Reflexive Symmetric and Transitive None of the above
3. (8 points) Let R be the relation defined on Z as follows: For every m,n ez, m Rn4|(m– n). Is R an equivalence relation? If so, prove it. If not, explain which properties of an equivalence relation fails by providing a counterexample for each property that is not satisfied.
: Let a1, a2, a3, . . . be the sequence of integers defined by a1 = 1 and defined for n ≥ 2 by the recurrence relation an = 3an−1 + 1. Using the Principle of Mathematical Induction, prove for all integers n ≥ 1 that an = (3 n − 1) /2 .
Let R be a relation defined on the integers Z by a R b if 6b^3 - 6a^3 <= 0 Which of the properties reflexive, symmetric, and transitive does R possess?
16. (8 points) Let Z be the integers and let A - Zx Z. Define the relation R on A by (a, b) R(c, d) if and only if a c and b 3 d for all (a, b), (c, d)E A. Prove that R is a partial ordering on A that is not a total ordering. 16. (8 points) Let Z be the integers and let A - Zx Z. Define the relation R on A by (a, b)...
Let G = {1, 3, 5, 9, 11, 13} and let represent the binary operation of multiplication modulo 14. (a) Prove that (G, ) is a group. (You may assume that multiplication is associative.) (b) List the cyclic subgroups of (G, ). (c) Explain why (G, ) is not isomorphic to the symmetric group S3. (d) State an isomorphism between (G, ) and (Z6, +).