2.4.3: Proving algebraic statements with direct proofs. Prove each of the following statements using a direct proof. a) for any positive real numbers, x and y, (x+y)^2 >= xy
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Prove or disprove each of the following. Note one of the statements is true. You must prove it. One of the statements is false. Prove it is false (with a counterexample). (a) Let x, y ∈ R. If x 2 < y2 , then x < y. (b) Let x and y be integers. If x and y are positive, then x + y ≥ 2 √xy
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Complete the following direct proof of the statement: If x and y are positive real numbers and x < y, then x^2 < y^2. Proof: Since x > 0, multiplying both sides of x < y by x yields x^2 < xy. Since x < y, multiplying both sides of x < y by y yields xy < y^2. Therefore, combining the two inequalities, we have x^2 < xy < y^2. Thus, we have proven that if x and y are positive real numbers and x < y, then x^2 < y^2.
Exercise 2.8.4: Proof by contrapositive of algebraic statements Prove each statement by contrapositive: For every pair of real numbers x and y, if x + y > 20, then x > 10 or y > 10. For every pair of positive real numbers x and y, if xy > 400, then x > 20 or y > 20.
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