[SOLVED] CS201 Endsem

Question 1.  Consider the following argument:

Let U be the set of all sets. Define a partial ordering on U by inclusion: A ≤ B iff A ⊆ B for A,B ∈ U. Consider a chain C of U under this partial ordering: C : A₁ ≤ A₂ ≤ A₃ ≤ ···. Define B = ∪_i≥1A_i. Clearly, B ∈ U and it is an upper bound of the chain C. Hence, Zorn’s Lemma implies that U has a maximal element, say M.

The argument is clearly wrong since M is not a maximal element: M ⊂ {M,{M}} ∈ U. Identify which step in the argument is wrong and why.

Question 2.  Let (G,·) be a finite group with the property that there exists only one element a₂ ∈ G such that a₂ 6= e and . Define a bipartite graph

H = (G,G,E) as follows.

Edge (a,b) ∈ E if a 6= e and b = a^k for k > 1, or a = e and b = a₂.

Prove that the graph H has a perfect matching.

Question 3.  Let R be a ring and a ∈ R. Define (a) = {b·a | b ∈ R}. Prove that (a) is an ideal of R.

Let polynomial C(x,y) = (x² +y² −1)·x. The curve C(x,y) = 0 is unit circle plus y-axis on the plane. (C) = {Q(x,y) · C(x,y) | Q(x,y) ∈ R[x,y]} is an ideal of the ring R[x,y], the ring of polynomials in two variables with coefficients in R.

Define R = R[x,y]/(C). For any point P ∈ R × R on the plane, define

R and g(P) 6= 0} and  and g(P) 6= 0 and f(P) = 0}. Prove that

R_P is a ring.

I_P is a maximal ideal of R_P.

For point P = (1,0), I_P = (y).

For point P = (0,1), (x) ⊆ I_P.

It can be shown that I_P 6= (x). Therefore, ring R_P contains information about whether curve C is degenerate at point P.

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