7.3 Isomorphisms of Interval-Valued Fuzzy GraphsIn this section, we consider various types of (weak) isomorphisms of interval-valuedfuzzy graphs.Definition 7.3.1 Let G₁ = (A₁, B₁) and G₂=(A2, B₂) be two interval-valuedfuzzy graphs. A homomorphism f : G₁ → G₂ is a mapping f: V₁ → V₂ suchthat for all x₁ € V₁, X₁y1 € E₁,+(i) μA, (x) ≤₂(f(x₁)), pt, (x₁) ≤ μ₂ (f(x₁)),(ii) µß, (x1Y₁) ≤ µB₂ (ƒ (x₁) ƒ (y₁)), µg, (x₁y₁) ≤ µg₂ (ƒ (x₁) ƒ (y₁)).A bijective homomorphism with the property(iii) μA, (x1) = μA₂ (ƒ (x₁)), μ₁ (x₁) = µ₂ (f(x₁))is called a weak isomorphism and a weak co-isomorphism if(iv) HB₁ (X1Y1) = HB₂ (f(x₁) ƒ (y₁)), PT, (x₁v₁) = µ/₂2 (f(x₁)ƒ (₁)) for allX1, y1 € V₁.A bijective mapping f: G₁ → G₂ satisfying (iii) and (iv) is called an isomor-phism.Now I want more examples of the subjectIsomorphisms of interval-Valued fuzzy graphs

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Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.5: Isomorphisms

Problem 20E: For each a in the group G, define a mapping ta:GG by ta(x)=axa1. Prove that ta is an automorphism of...

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Let A be a given nonempty set. As noted in Example 2 of section 3.1, S(A) is a group with respect to mapping composition. For a fixed element a In A, let Ha denote the set of all fS(A) such that f(a)=a.Prove that Ha is a subgroup of S(A). From Example 2 of section 3.1: Set A is a one to one mapping from A onto A and S(A) denotes the set of all permutations on A. S(A) is closed with respect to binary operation of mapping composition. The identity mapping I(A) in S(A), fIA=f=IAf for all fS(A), and also that each fS(A) has an inverse in S(A). Thus we conclude that S(A) is a group with respect to composition of mapping.
Find subgroups H and K of the group S(A) in example 3 of section 3.1 such that HK is not a subgroup of S(A). From Example 3 of section 3.1: A=1,2,3 and S(A) is a set of all permutations defined on A.
In Example 3, the group S(A) is nonabelian where A={ 1,2,3 }. Exhibit a set A such that S(A) is abelian. Example 3. We shall take A={ 1,2,3 } and obtain an explicit example of S(A). In order to define an element f of S(A), we need to specify f(1), f(2), and f(3). There are three possible choices for f(1). Since f is to be bijective, there are two choices for f(2) after f(1) has been designated, and then only once choice for f(3). Hence there are 3!=321 different mappings f in S(A).
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For each a in the group G, define a mapping ta:GG by ta(x)=axa1. Prove that ta is an automorphism of G. Sec. 4.6,32 Let a be a fixed element of the group G. According to Exercise 20 of Section 3.5, the mapping ta:GG defined by ta(x)=axa1 is an automorphism of G. Each of these automorphisms ta is called an inner automorphism of G. Prove that the set Inn(G)=taaG forms a normal subgroup of the group of all automorphisms of G.
True or False Label each of the following statements as either true or false. 12. The mutually disjoint cycles of a permutation are the same as its orbits.
An independent set of a graph G is a subset I of the vertex set V such that no twovertices in I are adjacent. Let i(G) be the size of a maximal independent set of G.(a) Show that I is an independent set of G if and only if V − I is a vertex coverof G.(b) Conclude from part (a) that i(G) + vc(G) = |V |.
If n<m then every linear map T: IRn -> IRm is one-to-one. Is this statement true or false?
Let G ={[1 0 0 1] ,[−1 0 0 1] ,[1 0 0 −1], [−1 0 0 −1]} . Is G ∼= K4? If yes, givean explicit isomorphism?
Consider maps π1 : G1 × G2 → G1 given by π1(g1, g2) = g1 and π2 : G1 × G2 → G2 given byπ2(g1, g2) = g2. Show that π1 and π2 are homomorphisms (You can just show one of them is a homomorphism and say the other one follows similarly.)

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