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  3. 2 use the following outline to supply proofs for the...

Question: 2 use the following outline to supply proofs for the...

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2. Use the following outline to supply proofs for the statements in Theorem 1 .4.13 (a) First prove statement () for two countable sets A and A. (See example 1.4.8 maybe). Some technicalities can be avoided by replacing A2 by B2 = {r E A2 : r A. The point of this is that the union A, U Ag is equal to Ar U B2, but A and B2 have no elements in common. (Dont forget B, might be finite). (b) Explain why induction cannot be used to prove (ii) of theorem 1.4.13 using (i). (c) Show how arranging N into a two-dimensional array 1 3 6 10 15 2 5 9 14.. 4 8 13. 7 12 . leads to a proof of theorem 1.4.13 (ii) (a) Prove that for all sets A, we have A~ A (b) Given sets A and B, prove that A~ B is equivalent to B~A (c) For three sets, A, B, and C, prove that if A~B and BC, then A~C. (Noto 3. that these three properties together prove that~is an equivalence relation) 4. Prove that the set of finite subsets of N is a countable set. 5. A real number z E R is called algebraic if there exists integers ao, a1,...,an E Z (not all zero), such that Said another way, a real number is algebraic if it is the root of a polynomial with integer coefficients. Real numbers that are not algebraic are called trunscendental numbers. Note that we dont know for sure right now that transcendental exist! Well soon see that they do :-)

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