Question

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.

Answer 1

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