Q1. Determine the properties of an equivalence relation that the others lack. This is an equivalence relation. The identity relation on set E is the set {(x, x) | x â E}. Any relation that can be expressed using \have the same" are \are the same" is an equivalence relation. 2. c) f(f;g)jf(x) g(x) = 1 8x 2Zg Answer: Re exive: NO f(x) f(x) = 0 6= 1. 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. The identity relation is true for all pairs whose first and second element are identical. Examples. 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. For each of these relations on the set {1,2,3,4}, decide whether it is reflexive, whether it is symmetric, whether is it antisymmetric, and whether is it transitive. Which of these relations on the set of all functions on Z !Z are equivalence relations? CCN2241 Discrete Structures Tutorial 6 Relations Exercise 9.1 (p. 527) 3. Another way to approach this is to try to partition people based on the relation. 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). b. 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¦ Which of these relations on the set of all functions from Z to Z are equivalence relations? Powers of a Relation Let R be a relation on the set A. First, reflexivity, symmetry, and transitivity of a relation requires that the properties are true for all elements of the set in question. Recall the following definitions: Let be a set and be a relation on the set . Which of these relations on the set of all people are equivalence relations? The objective is to tell for each of the following relations defined on the above set is reflexive, symmetric, anti-symmetric, transitive or not. For part B, you can part consider all pairs of people in the population: All these relations are definitions of the relation "likes" on the set {Ann, Bob, Chip}. \a and b are the same age." 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. 4 points a) 1 1 1 0 1 1 1 1 1 Consider the set as,. Reflexive relation: A relation is called reflexive relation if for every . The powers Rn;n = 1;2;3;:::, are deï¬ned recursively by R1 = R and Rn+1 = Rn R. 9.1 pg. Suppose A is a set and R is an equivalence relation on A. For each of these relations Thus R is an equivalence relation. 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