Prove, without using Theorem 1.12 or copying the proof, that for all a, b ∈ Q with a < b, there exists a c ∈ Q with a < c < b

Q: Prove that if alb and alc then a (bx + cy) for all integers x, y.

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Q: Prove that (ab + cd)² ≤ (a² + c²)(b² + ď²) for all a, b, c, d ɛ R.

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Q: Prove |°z 'z|>|°z|-|'z|

A: We have to prove  z1-z2≤z1-z2            orz1-z2≥z1-z2 NOTE : The polar form of a complex number…

Q: Prove that z =z iff z is real.

A: z represents a complex number which is in the form x+iy, where x and y are the real numbers and i is…

Q: Let n > 3 where x, y , z € R", |x – y| = d > 0, and r is some positive real number. Prove that if 2r…

A: It is given that x and y are d distance apart and r&gt;d/2 where x, y, z belongs to Rn So locus z…

Q: (b) Let a.. and m be positive integers. Prove that if (a,m) = (b, m) = 1, then (ab, m) = 1.

A: Given  a, b and m are positive integers.  (a, m) = (b, m) = 1.We have to prove that (ab, m) = 1.

Q: Let x, y EZ. Prove that 3|xy if and only if 3|x x or 3 y.

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Q: Let a ∈ Z and b ∈ N. Prove that there exists unique q, r ∈ Z such that a = bq + r and 0 ≤ r < b

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Q: Prove that ∀a, b, k ∈ Z that ((4|k) ∧ (k = ab)) ⇒ (2|a ∨ 2|b)

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Q: Show that if a|m and a|(ms+nt) for some integers a≠0, m, s, n, and t, then a|nt.

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Q: Let a, b € Z with a | b. Prove that an | bn for every positive integer n.

A: Given that           a, b∈ℤ with a|b. To prove an|bn  , we shall use mathematical induction.

Q: Prove that gcd(a, b) is the least natural number representable in the form ax + by with x, y ∈ Z.

A: We have to show that gcd(a, b) is the least natural number representable in the form ax + by with x,…

Q: Prove that R(5, 5) < 50.

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Q: Prove that for any set ACR and any real number a we have m* (A +a)= m* (A).

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Q: 3. Prove that if a 0.

A: To prove that if a&lt;0 and b&lt;0, then ab&gt;0.

Q: Prove that if x is not free in the formula A, then ` A ≡ (∃x)A

A: The formula is -------(1). To Prove: x is not free in the formula Proof: Let us assume that A(x) :…

Q: Prove that there do not exist a, b ∈ Z satisfying 54a − 39b = 2.

A: Solution:

Q: Let a, b, c, d, e, and f be positive integers. Let d = gcd(a, b, c), e = gcd(a, b) and f = gcd(e,…

A: Given that for positive integers a,b,c,d,e,f , d=gcda,b,ce=gcda,bf=gcde,c

Q: If a € R such that 0 0, then show that a = 0.

A:

Q: Prove that exists integers x, y, and z such that 1/x + 1/y + 1/z is also an integer.

A: Given: The expression is 1x+1y+1z where x, y and z are integers. To prove: There exist integers x,…

Q: Prove that:  If 0 ≠ p ∈ Q and x ∉ Q, then (show that) a + x ∉ Q and ax ∉ q.

A: Given: Let a∈ℚ and x∉ℚ To show: a+x∉ℚax∉ℚ

Q: Let a and b be any real numbers such that a 0. Prove that a < b.

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Q: Let S be a set of real numbers such that 5 ∈ S and the sup S > 5. Is it required that S has to have…

A: Let S be a set of real numbers such that 5∈S and thesupS&gt;5.Is it required that S has to have a…

Q: Show that: for any z, w in C. ||2|-|w||≤|zw|≤|z| + |w|

A: we have to show that  ≤ ≤ for any z and w in .

Q: Prove the following statement using the axioms and lemmas For all nonzero real numbers w, x and y,…

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Q: Let A  = {x ∈ A  |x =3b +2 for some integer b } and B = {y ∈ B |y = a + 2 for some integer a }  a.…

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Q: "for all n,m ∈Z \ {0} and g ∈ N,  gcd (n,m)= g then there are a,b ∈ Z with an+bm= g" is the…

A: We need to see if the converse of the statement "gcdn,m=g then there are a,b∈Z with an+bm=g" is true…

Q: How do we prove the following by doing a direct proof? " Let a  ∈  Z. Prove that if 3|2a, then 3|a."

A: This question is done by the help of number theory concept.

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.