Answer: Assign a grade of A (correct), C (partially correct), or F (failure) to

Chapter 3, Problem 19

(choose chapter or problem)

Assign a grade of A (correct), C (partially correct), or F (failure) to each.Justify assignments of grades other than A.(a) Claim. If the relation R is symmetric and transitive, it is also reflexive.Proof. Since R is symmetric, if then Thusand and since R is transitive, Therefore,R is reflexive.(b) Claim. The relation T on given byr + s is symmetric. (x, y) T (r, s) iff x + y =(x, y) R (y, x) R, (x, x) R.(x, y) R, (y, x) R.DEFINITION Let A be a nonempty set. is a partition of A isa set of subsets of A such that(i) If then(ii) If and then or(iii) XX = A.X Y , X = Y X Y = .X , X = . iffProof. Suppose Then becauseTherefore, T is symmetric.(c) Claim. The relation W on given byis symmetric.Proof. Suppose and are in andThen Therefore, so ThusW is symmetric.(d) Claim. If the relations R and S are symmetric, then is symmetric.Proof. Let R be the relation of congruence modulo 10 and S therelation of congruence modulo 6 on the integers. Both R and S are symmetric.If then 6 and 10 divide Therefore, 2, 3, and5 all divide so 30 divides Also if 30 divides then 6and 10 divide so is the relation of congruence modulo 30.Therefore, is symmetric.(e) Claim. If the relations R and S are symmetric, then is symmetric.Proof. Suppose Then and Since Rand S are symmetric, and Therefore, (f) Claim. If the relations R and S are transitive, then is transitive.Proof. Suppose and Thenand Therefore,3.3 PartitionsPartiti(x, z) R S.