# You're the grader. To each "Proof", assign one of the following grades: A (correct), if the claim and proof are correct, even if the proof is not the simplest, or the proof you would…

You're the grader. To each "Proof", assign one of the following grades: A (correct), if the claim and proof are correct, even if the proof is not the simplest, or the proof you would have given. C (partially correct), if the claim is correct and the proof is largely a correct claim, but contains one or two incorrect statements or justifications. . F (failure), if the claim is incorrect, the main idea of the proof is incorrect, or most of the statements in the proof are incorrect. Give a brief justification for grades of C or P. (PI) Claim. The following relation defined on R2 is symmetric: (α, b) ~ (x, y) a+b x+y. "Proof." Suppose (a, b) E R2. Then (a,b) ~ (b, a) because a+b is symmetric. (P2) Claim. The following relation defined on R2 is symmetric: b+a. Therefore, "Proof," Supposete, y) ~ (r, s). Then z-r (r,s)~(x, y). Thus ~is symmetrio. Thefore, r-z = s-y, so y-s. (P3) Claim. Let ~be an equivalence relation on X and and let z, y,z e X. If z Ey and z f Eg, then z fEy Proof." Assume that r e E, and z e E. Then y ~ r and z, so by transitivity, y ~ z. Therefore, if z e E, and z E, it must be that z EV. (P4) Claim. Let ~be an equivalence relation on X and and let z, y,z e X. If r E E and z B,, then E, Proof." Assume that r e Ey and z e Ey. Then y~z and y z. By symmetry a ~y, and by transitivity, a~z. Therefore, z e E. We conclude that if z E Ey and z E, then z Ey. (P5) Claim. If A is a partition of A and 1B is a partition of B, then AUB is a partition of AUB. "Proof" (i) ICCAUB, then C A or C E B. In either case, C o. (ii) Since A-UCeA C and B UDe D it follows that AUB -UBEAU (ii) If CE AuB and D E AUB, then we have four cases: C, D e A or C, D E B or (C E A and D E B) or (C B and D A). However, since A and B are both partitions, we have either C- D or CnD- in each case.