10 Required informationNOTE This is a multi-part question Once an answer is submitted, you will be unable to return to this partFor each of these relations on the set (1, 2, 3, 4), decide whether it is reflexive, symmetric, antisymmetric, or transitive(1. 1. (1, 2), (2. 9. (2 2), (3, 3), (4. 4) (Check all that apply.)Check All Thet Applytransitivesymmietricantisymmetricrefiexivenone of these properties

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Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 6TFE: Label each of the following statements as either true or false. Let R be a relation on a nonempty...

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True or False Label each of the following statements as either true or false. The set ZZ+ is closed with respect to subtraction.
6. a. Give an example of mappings and , different from those in Example , where is one-to-one, is onto, and is not one-to-one. b. Give an example of mappings and , different from Example , where is one-to-one, is onto, and is not onto.
Label each of the following statements as either true or false. Let R be a relation on a nonempty set A that is symmetric and transitive. Since R is symmetric xRy implies yRx. Since R is transitive xRy and yRx implies xRx. Hence R is alsoreflexive and thus an equivalence relation on A.
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.
PLEASE TYPE ONLY IF NOT TYPE THAN DO NOT DO IT*** PLEASE DO ALL TWO*** Exercise 8.4.5: Composition and relation properties. For the following statements, provide a proof if the statement is true or give a counterexample if the statement is false. S and R are binary relations over the same domain. (e) If S and R are both anti-symmetric, then S∘R is anti-symmetric. (f) If S is anti-symmetric, then S∘S is anti-symmetric.
1. Let R and S be transitive relations on a non-empty set A. Prove or disprove the following: a) if R2 ⊆ R then R is transitive. b) if S ∘ R ⊆ R ∘ S then R ∘ S is transitive.
H3. Given set A = {1, 2, 3, 4}, mention 5 or more relations (R) defined on A such that R does not satisfy any of the properties (reflexive, irreflexive, symmetric, asymmetric, antisymmetric, transitive, equivalence, partial order, total order, well order).
True or False?11. If Γ = {[0, 1/n) | n ∈ N} then ∩Γ = [0, 1).12. Let R be a relation on a set A. Then R is symmetric if and only if R^(−1) ⊆ R.
Let R be the relation on the set of all mathematicians that contains the ordered pair (a, b) if and only if a and b have written a published mathematical paper together. a) Describe the relation R[2] in the context of the problem.b) Describe the relation R* in the context of the problem.c) The Erdos number of a mathematician is 1 if this mathematician wrote a paper with the prolific Hungarian mathematician Paul Erdos. The Erdos number is 2 if a mathematician didn’t write a joint paper with Erdos but wrote a joint paper with someone who wrote a joint paper with Erdos, and so on (except that the Erdos number of Erdos himself is 0). Give a definition of the Erdos number is terms of paths in R. List a name of a mathematician with number: 1, 2, and 3
1. Let A = {2,4,6}, let S = P(A) and let R ⊆ S x S be a relation defined by the rule (X,Y) ∈ R ⇔ (X ∩ Y = Ø) ∧ (X ∪ Y = A). (a) Prove or find a counterexample for the following properties and based on the findings from the properties, conclude whether R is a graph, partially ordered set, equivalence relation, or a function (several answers may hold simultaneously). Analyze the relation R.: antisymmetry asymmetry symmetry irreflexivity reflexivity transitivity uniqueness
Question: Construct a smallest binary relation S defined on the set {w, x, y, z} such that S satisfies all of the following requirements: S contains (x, y). S is not transitive. S is not antisymmetric. S is not symmetric. |S| is even.
(1) The relation △ means “have a common factor that isn’t 1”. For example, 6 △ 4 because they both are divisible by 2, and 10 △ 15 because they are both divisible by 5. Draw a digraph of △ on {2, 3, 4, 5, 6, 7, 8, 9}. (2) Is the relation △ . . . (a) Transitive? (b) Reflexive? (c) Antisymmetric? (d) Symmetric? (e) Equivalence relation? (f) Partial ordering?

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