Discrete Math 3. Let f AB and g B C be functions. Prove that if gof is onto and g isone-to-one, then f is onto.4. Show that the open intervals (-1,2) and (0, ∞0) have the same cardinality.

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Answered: 3. Let f AB and g B C be functions.…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 20E: Give an example of a relation R on a nonempty set A that is symmetric and transitive, but not...

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Similar questions

Give an example of a relation R on a nonempty set A that is symmetric and transitive, but not reflexive.
Let f and g be permutations on A. Prove that (fg)1=g1f1.
Determine whether the set S={1,x2,2+x2} spans P2.
Prove that if f is a permutation on A, then (f1)1=f.
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.
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.
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.
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.
7. a. Give an example of mappings and , where is onto, is one-to-one, and is not one-to-one. b. Give an example of mappings and , different from example , where is onto, is one-to-one, and is not onto.

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3. Let f AB and g B C be functions. Prove that if gof is onto and g is one-to-one, then f is onto. 4. Show that the open intervals (-1,2) and (0, ∞0) have the same cardinality.