7.3 Isomorphisms of Interval-Valued Fuzzy Graphs In this section, we consider various types of (weak) isomorphisms of interval-valued fuzzy graphs. Definition 7.3.1 Let G₁ = (A₁, B₁) and G₂ = (A2, B₂) be two interval-valued fuzzy graphs. A homomorphism f: G₁ G₂ is a mapping f: V₁ → V₂ such that for all x₁ € V₁, X₁Y₁ € E₁, (i) μÃ, (x₁) ≤ PA₂ (f(x₁)), μ, (x₁) ≤ µ₂ (f(x₁)), (ii) PB, (X1Y1) ≤ PB₂ (ƒ (x₁) ƒ (y₁)), µg, (x₁ y₁) ≤ PB₂ (f(x₁) ƒ (y₁)). A bijective homomorphism with the property (iii) μ (x₁) = μ₂ (f(x₁)), µμ₁ (x₁) = µ₂ (f(x₂)) is called a weak isomorphism and a weak co-isomorphism if PB₂ (f(x₁) f (y₁)), P, (x₁3₁) = (iv) HB, (X1Y1) X₁, Y₁ € V₁. A bijective mapping f: G₁ G₂ satisfying (iii) and (iv) is called an isomor- phism. Example 7.3.3 Let G₁ and G₂ be as in Example 7.3.2 and let A₁, A₂, B₁, and B₂ b interval-valued fuzzy subsets defined by A₁ b₁ a1 - ((1₂²) 03). (₁ 0²5)). B₁ = = 0.2/ 0.4' A₂ = (f(x₁)f(y)) for all a2 ((04-03) (05.06)) B₂ = ab abi 0.1 0.3 | a2b2a2b2 0.1 0.3 Then G₁ = (A₁, B₁) and G₂ = (A2, B₂) are interval-valued fuzzy graphs of G and G₂, respectively. The map f: V₁ V₂ defined by f(a) = b₂ and f(b₁) = a₂ is weak co-isomorphism, but it is not an isomorphism. Now I want more examples of the subject Isomorphisms of interval-Valued fuzzy graphs
7.3 Isomorphisms of Interval-Valued Fuzzy Graphs In this section, we consider various types of (weak) isomorphisms of interval-valued fuzzy graphs. Definition 7.3.1 Let G₁ = (A₁, B₁) and G₂ = (A2, B₂) be two interval-valued fuzzy graphs. A homomorphism f: G₁ G₂ is a mapping f: V₁ → V₂ such that for all x₁ € V₁, X₁Y₁ € E₁, (i) μÃ, (x₁) ≤ PA₂ (f(x₁)), μ, (x₁) ≤ µ₂ (f(x₁)), (ii) PB, (X1Y1) ≤ PB₂ (ƒ (x₁) ƒ (y₁)), µg, (x₁ y₁) ≤ PB₂ (f(x₁) ƒ (y₁)). A bijective homomorphism with the property (iii) μ (x₁) = μ₂ (f(x₁)), µμ₁ (x₁) = µ₂ (f(x₂)) is called a weak isomorphism and a weak co-isomorphism if PB₂ (f(x₁) f (y₁)), P, (x₁3₁) = (iv) HB, (X1Y1) X₁, Y₁ € V₁. A bijective mapping f: G₁ G₂ satisfying (iii) and (iv) is called an isomor- phism. Example 7.3.3 Let G₁ and G₂ be as in Example 7.3.2 and let A₁, A₂, B₁, and B₂ b interval-valued fuzzy subsets defined by A₁ b₁ a1 - ((1₂²) 03). (₁ 0²5)). B₁ = = 0.2/ 0.4' A₂ = (f(x₁)f(y)) for all a2 ((04-03) (05.06)) B₂ = ab abi 0.1 0.3 | a2b2a2b2 0.1 0.3 Then G₁ = (A₁, B₁) and G₂ = (A2, B₂) are interval-valued fuzzy graphs of G and G₂, respectively. The map f: V₁ V₂ defined by f(a) = b₂ and f(b₁) = a₂ is weak co-isomorphism, but it is not an isomorphism. Now I want more examples of the subject Isomorphisms of interval-Valued fuzzy graphs
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.5: Graphs Of Functions
Problem 44E
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