MHF 4102 - AX. SET THEORY FLORIDA INT'L UNIV.

TEST #1 - Sprinq '07 TIME: 75 min.

Answer all 6 questions. Provide all reasoning and show allworking. An unjustified answer will receive little or no credit.Begin each question on a separate page.

(15) 1. Let <Ai: iEI) be an indexed family of subsets of the set X.Prove that

u (X-AJiEI

X - (n Ai ).iEI

(15) 2 Let f: X -+ Y be a function and suppose that A, B ~X and C, D ~Y.

(a) Is it always true that f[A]nf[B] ~ f[AnB] ?(b) Is it always true that f-1[C]nf-1[D] ~ f-1[C n D] ?

(20) 3 (a) Write down the Separation and Replacement axioms in both theirordinary form and in class-form.

(b) Write down the Union and Fqundation axioms in their ordinaryform. Then translate them completely into the language of settheory.

(15)4 (a) Using the fact that R = {x: x~x} is not a set, prove that V ={x: x=x} is not a set.

(b) Prove that the collection S2 of all sets with exactly twoelement is a proper class.

(15) 5(a) Define what is a minimal element and what is a minimum elementof a partially ordered set (A,<).

(b) Define what it means for (A,<) to be well-founded. Using theFoundation Axiom prove that there is no set x such that XEx.

(20) 6(a) Define ordinal addition and ordinal multiplication.(b) Prove that if f)< y, then cx+f)< cx+y for all ordinals

transfinite induction.

(c) Hence show that if cx+f)=cx+y,then f)=y.

cx,f),y by

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