علاقه - Conversely if there exist an operation on P(x) satisfying K, to ky then there exist a unique topology I on x such that (A) = CL(A) for each AEP(x)-
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- Prove Theorem 1.40: If is an equivalence relation on the nonempty set , then the distinct equivalence classes of form a partition of .A relation R on a nonempty set A is called asymmetric if, for x and y in A, xRy implies yRx. Which of the relations in Exercise 2 areasymmetric? In each of the following parts, a relation R is defined on the set of all integers. Determine in each case whether or not R is reflexive, symmetric, or transitive. Justify your answers. a. xRy if and only if x=2y. b. xRy if and only if x=y. c. xRy if and only if y=xk for some k in . d. xRy if and only if xy. e. xRy if and only if xy. f. xRy if and only if x=|y|. g. xRy if and only if |x||y+1|. h. xRy if and only if xy i. xRy if and only if xy j. xRy if and only if |xy|=1. k. xRy if and only if |xy|1.Let be a relation defined on the set of all integers by if and only if sum of and is odd. Decide whether or not is an equivalence relation. Justify your decision.
- 3bLet R, S, and T be sets. Let f: R -» S, and g: S -» T be maps. Assume we know that qf is 1-1 Must f be 1-1? Either prove that it is or find a counterexample (a) Must g be 1-1? Either prove that it is or find a counterexample (b)1. (a) (b) (c) (d) Prove or disprove that, for any universal set U and predicates P and Q, [3x = U, P(x) ^ Q(x)] → [3r EU, P(x)) ^ (3x = U, Q(x))] Prove or disprove that, for any universal set U and predicates P and Q, [3r EU, P(x)) ^ (3xU, Q(x))] → [r U, P(x) ^ Q(x)] Prove or disprove that, for any universal set U and predicate P [3r € U, P(x)] → [Vr € U, P(x)] Prove or disprove that, for any universal set U and predicate P [VxU, P(x)] → [3r € U, P(x)]
- 4. Define a relation R on NN by aRb if a and b have no common factors except 1. (a) ( Give an example of a pair (a, b) with (a, b) € R. (b) Give an example of a pair (a, b) with (a, b) & R.Find Let f : R² → R² be an isomorphism where (C)--(4)-1] [8]- Let (-) be the evaluation inner product on P₂ at -1, 0, and 1. That is, for any p = p(x) and q = q (x), we have (p, q) = p(-1)q(-1) +p(0)q(0) +p(1)q(1). Determine whether or not the set {1+x², x, x + x²} is orthogonal in this inner product.