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Math 300 - Introduction to Mathematical reasoning Summer 2013 Practice Midterm 1 1. Suppose you have two statements P (x, y) : Q(x, y) : x and y are two integers such that x + y is even. x and y are two integers such that x and y are odd. (a) Only one of these statements implies the other statement. Does P =⇒ Q or does Q =⇒ P ? (No proof required.) (b) Give a concrete example to show that reverse implication does not hold (No Proof required). 2. Let A and B be two sets. Prove that A ∪ B = A ∪ (B − A). 3. Let f : X → Y and g : Y → X be two functions such that the composition g ◦ f is the identity function (i.e. if x ∈ X, then (g ◦ f )(x) = x). Prove the following : (a) f is injective. (b) g is surjective. √ 4. In this question, we shall prove that 3 is not a rational number. Please use the hints provided (and NOT another technique). 2 (a) In class, we have used √ the fact that f (x) = x is an increasing function. Use this to show that 3/2 < 3 < 2. √ √ (b) Suppose that 3 was a positive rational number. Let us say that 3 is equal to c/d, where c and d are natural numbers. Use part 4(a) to show that 3d − c < c. √ (c) Use the well-ordering principle to conclude that 3 is not a rational number. 1-1 Lecture 2: 2-1 Math 300 - Introduction to Mathematical reasoning Summer 2013 Practice Midterm 2 1. Negate the following statement without the using the word “NOT” “The sequence {an } is bounded.” 2. Use induction to prove the following formula 12 + 22 + 32 + · · · + n2 = n(n + 1)(2n + 1) . 6 3. Let P , Q and R be three logical statements. Two of the following three statements are logically equivalent to each other. (P =⇒ Q) =⇒ R P =⇒ (Q =⇒ R) (P =⇒ R) ∨ (P =⇒ ¬Q) (a) Identify the two logically equivalent statements. Prove that they are logically equivalent. (b) Why is the remaining statement NOT logically equivalent to the two logically equivalent statements in Part 4(a) ? (You must provide a concrete example where this statement is not logically equivalent to the two other statements.) 4. Give a concrete function f : Z → N that is injective but NOT surjective. Specify a natural number which is not contained in the image of f . (You do not have to provide proofs. However, you must define your function f properly.)