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) ≤μÃ₂ (f (x₁)), μ, (x₁) ≤ μ₂ (f(x₁)), + (ii) μg, (x1Y1) ≤ PB₂ (f(x₁) ƒ (y₁)), µg, (x₁y₁) ≤ ₂ (f(x₁) f (y)). A bijective homomorphism with the property (iii) μA, (x₁) = μÃ₂ (f(x₁)), μ₁ (x₁) = μ₂ (f(x₁)) is called a weak isomorphism and a weak co-isomorphism if = (iv) B₁ (x1y₁) = PB₂ (f(x₁) f (y1₁)), P, (X1Y₁) X1, y1 € V₁. A bijective mapping f: G₁ G₂ satisfying (iii) and (iv) is called an isomor- phism. (f(x₁) f(y)) for all

11 Part 3 of 3 Required information NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. For each of these relations on the set (1, 2, 3, 4), decide whether it is reflexive, symmetric, antisymmetric, or transitive. {(1, 3), (1. 4), (2, 3), (2, 4), (3, 1), (3, 4)] (Click to select) ✓ (Click to select) antisymmetric none of these transitive reflexive symmetric

coloring graph theory

14 Part 1 of 4 Required information NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Consider the following relations: R₁ = {(a, b) = R² | a> b), the greater than relation R₂ = {(a, b) € R2 R3 = {(a, b) € R² | a < b), the less than relation R₁ = [(a, b) ER² | a ≤ b), the less than or equal to relation R5 = {(a, b) € R² | a= b), the equal to relation R₁ = {(a, b) = R² | a b), the unequal to relation For these relations on the set of real numbers, find R₂0 R2. Multiple Choice R4 R3 I az b), the greater than or equal to relation R2 R₁