MAT 300 WRITING ASSIGNMENT 2 NAME: (a) Critique the following suggested proofs. Indicate all the errors (if there is any), as you would grade a homework in details on a 6 point scale.

MAT 300 WRITING ASSIGNMENT 2 NAME: (a) Critique the following suggested proofs. Indicate all the errors (if there is any), as you would grade a homework in details on a 6 point scale. Indicate the mistakes between the lines. In this part you do not need to fix the proof, only comment on the errors and assign a grade. (b) On the other side of the paper write out clearly a correct proof for the one(s) that were not correct. It can look different then the one provided below. Example 1 Suppose A \ B ⊆ C ∩ D and x ∈ A. Prove that if x /∈ C then x ∈ B . Suggested proof: Suppose x ∈ A and x /∈ B. Then, by definition of difference, x ∈ A \ B. By definition of subset, for all x, if x ∈ A \ B then x ∈ C ∩ D. By definition of intersection x ∈ C ∧ D. Thus, x ∈ C. Example 2 Prove that there exist two irrational numbers a and b such that a b is rational. Suggested proof: We know that √ 2 is irrational. Consider c = √ 2 √ 2 . Case 1: c is rational. Then let a = √ 2 and b = √ 2. Then clearly, a b is rational. Case 2: c is irrational. Then let a = √ 2 √ 2 and b = √ 2. Then a b = √ 2 √ 2 √ 2 = √ 2 2 = 2. Thus, we have proved that there exist two irrational numbers a and b such that a b is rational. Example 3 Suppose F = {Ai | i ∈ Z +} is a family of infinitely many sets. Prove that if there are infinitely many positive integers i such that x ∈ Ai then x ∈ \∞ n=1 [∞ i=n Ai ! . Suggested proof: Suppose that x ∈ Ai for all i ∈ Z +. Then x ∈ S∞ i=n Ai for all n. By the definition of intersection, we can conclude that x ∈ \∞ n=1 [∞ i=n Ai ! . 1

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