Let a be a positive element in an ordered field. Show that if n is an...

Let a be a positive element in an ordered field. Show that if n is an odd number, a has at most one nth root; if n is an even number, a has at most two nth roots.

Expert Solution

Colby Messinger answered 2 years ago

The question is correct. Let X be an n-element set of positive integers each of whose...

The question is correct. Let X be an n-element set of positive integers each of whose elements is at most (2n - 2)/n. Use the pigeonhole principle to show that X has 2 distinct nonempty subsets A ≠ B with the property that the sum of the elements in A is equal to the sum of the elements in B.

Let (F, <) be an ordered field, let S be a nonempty subset of F, let...

Let (F, <) be an ordered field, let S be a nonempty subset of F, let c ∈ F, and for purposes of this problem let cS = {cx | x ∈ S}. (Do not use this notation outside this problem without defining what you mean by the notation.) Assume that c > 0. (i) Show that an element b ∈ F is an upper bound for S if and only if cb is an upper bound for cS. (ii)...

Let (Z, N, +, ·) be an ordered integral domain. Let {x1, x2, . . ....

Let (Z, N, +, ·) be an ordered integral domain. Let {x1, x2, . . . , xn} be a subset of Z. Prove there exists an i, 1 ≤ i ≤ n such that xi ≥ xj for all 1 ≤ j ≤ n. Prove that Z is an infinite set. (Remark: How do you tell if a set is infinite??)

Let A ∈ Rn×n be symmetric and positive definite and let C ∈ Rn×m. Show:, rank(C^TAC)...

Let A ∈ Rn×n be symmetric and positive definite and let C ∈ Rn×m. Show:, rank(C^TAC) = rank(C),

Let n be a positive integer. Prove that if n is composite, then n has a...

Let n be a positive integer. Prove that if n is composite, then n has a prime factor less than or equal to sqrt(n) . (Hint: first show that n has a factor less than or equal to sqrt(n) )

Let n be a positive integer. Let S(n) = n sigma j=1 ((1/3j − 2) −...

Let n be a positive integer. Let S(n) = n sigma j=1 ((1/3j − 2) − (1/3j + 1)). a) Compute the value of S(1), S(2), S(3), and S(4). b) Make a conjecture that gives a closed form (i.e., not a summation) formula for the value of S(n). c) Use induction to prove your conjecture is correct.

Let ?∈ℕ, and assume √? is irrational. Show that ℚ(√?)={?+?√?∶?,?∈ℚ} is a field (show that there...

Let ?∈ℕ, and assume √? is irrational. Show that ℚ(√?)={?+?√?∶?,?∈ℚ} is a field (show that there is multiplicative commutativity and multiplicative inverse). What would change if ℚ was replaced with ℝ.

Let p be an element in N, and define d _p to be the set of...

Let p be an element in N, and define d _p to be the set of all pairs (l,m) in N×N such that p divides m−l. Show that d_p is an equivalence relation

Let A be a commutative ring and F a field. Show that A is an algebra...

Let A be a commutative ring and F a field. Show that A is an algebra over F if and only if A contains (an isomorphic copy of) F as a subring.

Let F be an ordered field. We say that F has the Cauchy Completeness Property if...

Let F be an ordered field. We say that F has the Cauchy Completeness Property if every Cauchy sequence in F converges in F. Prove that the Cauchy Completeness Property and the Archimedean Property imply the Least Upper Bound Property. Recall: Least Upper Bound Property: Let F be an ordered field. F has the Least Upper Bound Property if every nonempty subset of F that is bounded above has a least upper bound.

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