notes Let A and B be sets. If there exists an injection f: of A is less cardinality aof B, written A ĥ B. If there exists á surjectionf:A→B, the cardinality of A is greater than or equal to the writfen A →Bithe cardinality than or equal to the cardină lity of B, al Prove that if A ĥ B, then B2 A. b) Prove that if A Ś B and Aà B, then AnB.

notes Let A and B be sets. If there exists an injection f: of A is less cardinality aof B, written A ĥ B. If there exists á surjectionf:A→B, the cardinality of A is greater than or equal to the writfen A →Bithe cardinality than or equal to the cardină lity of B, al Prove that if A ĥ B, then B2 A. b) Prove that if A Ś B and Aà B, then AnB.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 16E: 16. Let and define on by if and only if . Determine whether is reflexive, symmetric, or...

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13. Consider the set of all nonempty subsets of . Determine whether the given relation on is reflexive, symmetric or transitive. Justify your answers. a. if and only if is subset of . b. if and only if is a proper subset of . c. if and only if and have the same number of elements.
16. Let and define on by if and only if . Determine whether is reflexive, symmetric, or transitive.
21. A relation on a nonempty set is called irreflexive if for all. Which of the relations in Exercise 2 are irreflexive? 2. In each of the following parts, a relation is defined on the set of all integers. Determine in each case whether or not is reflexive, symmetric, or transitive. Justify your answers. a. if and only if b. if and only if c. if and only if for some in . d. if and only if e. if and only if f. if and only if g. if and only if h. if and only if i. if and only if j. if and only if. k. if and only if.
Label each of the following statements as either true or false. Let f:AB. Then f(A)=B for all nonempty sets A and B.
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.
Label each of the following statements as either true or false. 3. Let where A and B are nonempty. Then for every subset S of A.