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# exercises_5 - a b √ 2 c √ 3 = 0 Prove that a = b = c =...

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Analysis 1: Exercises 5 1. Prove that (a) ( n N ) [( n > 10) (2 n > n 3 )]. (b) ( n N ) [( n > 9) (2 n > 4 n 2 + 1)]. 2. The notation ‘( ! x S ) P ( x )’ is often used to mean that there is a unique element x of S for which P ( x ) is true (i.e., there is exactly one element for which P ( x ) is true). Write the statement using the usual quantiﬁers. 3. Let b, d, q be positive integers and c d < p q < a b . Show that one can ﬁnd positive integers m, n such that p q = ma + nc mb + nd . Construct a numerical example. 4. (a) Prove that no rational has its cube equal 16. (b) Prove that a rational number p/q in its lowest terms is the cube of a rational number iﬀ there exist integers m and n such that p = m 3 and q = n 3 . 5. (a) Let a, b, c be rational and
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Unformatted text preview: a + b √ 2 + c √ 3 = 0 . Prove that a = b = c = 0. (b) Let a, b, c, d be rational and a + √ b = c + √ d. Prove that either a = c and b = d or b and d are squares of rationals. (c) Let a, b, c, d be rational and x be irrational. In what circumstances is ax + b cx + d rational? 6. Using the axioms of real numbers show that (a) ( ∀ a ∈ R )( ∀ b ∈ R )( ∀ c ∈ R )[(( ac = ab ) ∧ ( a 6 = 0)) = ⇒ ( b = c )]; (b) ( ∀ a ∈ R )( ∀ b ∈ R )[(-a )(-b ) = ab ]; (c) ( ∀ a ∈ R )( ∀ b ∈ R )( ∀ c ∈ R )[(( a > b ) ∧ ( c < 0)) = ⇒ ( ac < bc )]....
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