Say whether each of the following is true or false, and support your decision by a proof: (a) There exist real numbers x and y such that x + y = y. (b) For all positive integers n, n" + n + 41 is a prime. (c) ?x?y(x + y = 0) (where x, y are real number variables). (d) (?m ? N)(?n ? N)(3m + 5n = 12). (e) For all integers a, b, c, if a divides bc (without remainder), then either a divides b or a divides c. (f) The sum of any five consecutive integers is divisible by 5 (without remainder). (g) For any integer n, the number n" + n + 1 is odd. (h) Between any two distinct rational numbers there is a third rational number. (i) For any real numbers x, y, if x is rational and y is irrational, then x + y is irrational. (j) For any real numbers x, y, if x + y is irrational, then at least one of x, y is irrational. (k) For any real numbers x, y, if x + y is rational, then at least one of x, y is rational.
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- Statement: There exist real numbers \( x \) and \( y \) such that \( x + y = y - Uhi6r \). - Simplify: \( x + y = y - Uhi6r \) implies \( x = -Uhi6r \). - Conclusion: True, as \( x = -Uhi6r \) is a real number. Show moreā¦
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