581 # 3 For each of these relations on the set f1;2;3;4g, decide whether it is reï¬exive, whether it is sym-metric, whether it is antisymmetric, and whether it is transitive. Suppose A is a set and R is an equivalence relation on A. \a and b are the same age." First, reflexivity, symmetry, and transitivity of a relation requires that the properties are true for all elements of the set in question. Determine the properties of an equivalence relation that the others lack. Powers of a Relation Let R be a relation on the set A. The objective is to tell for each of the following relations defined on the above set is reflexive, symmetric, anti-symmetric, transitive or not. Thus R is an equivalence relation. Recall the following definitions: Let be a set and be a relation on the set . So for part A, you can partition people into distinct sets: First set is all people aged 0; Second set is all people aged 1; Third set is all people aged 2; Etc. 4 points a) 1 1 1 0 1 1 1 1 1 a. Which of these relations on the set of all functions from Z to Z are equivalence relations? Any relation that can be expressed using \have the same" are \are the same" is an equivalence relation. The identity relation on set E is the set {(x, x) | x â E}. Hence ( f;f) is not in relation. CCN2241 Discrete Structures Tutorial 6 Relations Exercise 9.1 (p. 527) 3. You need to be careful, as was pointed out, with your phrasing of "can have" which implies "there exists", and your invocation of the $\leq$ relation to address problem (a). For each element a in A, the equivalence class of a, denoted [a] and called the class of a for short, is the set â¦ b. 14) Determine whether the relations represented by the following zero-one matrices are equivalence relations. Equivalence relations on a set and partial order Hot Network Questions Word for: "Repeatedly doing something you are scared of, in order to overcome that fear in time" Determine the properties of an equivalence relation that the others lack. All these relations are definitions of the relation "likes" on the set {Ann, Bob, Chip}. Examples. Q1. Happy world In this world, "likes" is the full relation on the universe. c) f(f;g)jf(x) g(x) = 1 8x 2Zg Answer: Re exive: NO f(x) f(x) = 0 6= 1. The identity relation is true for all pairs whose first and second element are identical. Another way to approach this is to try to partition people based on the relation. For part B, you can part consider all pairs of people in the population: we know that ad = bc, and cf = de, multiplying these two equations we get adcf = bcde => af = be => ((a, b), (e, f)) â R Hence it is transitive. 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