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I've tried this question with Bing Chat (GPT-4) and Bard: > A total order requires a <= relation which must be reflexive, transitive, anti-symmetric, and stron

I've tried this question with Bing Chat (GPT-4) and Bard: > A total order requires a <= relation which must be reflexive, transitive, anti-symmetric, and strongly connected. Give a definition for <= on real numbers defined using Cauchy sequences. This is the result Bing Chat gave me about a week ago (although I've just tried it a few times now, and all of the answers are considerably worse): 'A Cauchy sequence is a sequence of real numbers such that for any positive real number ε, there is a positive integer N such that for all natural numbers m and n greater than N, the absolute value of the difference between the m-th and n-th terms of the sequence is less than ε. One way to define a <= relation on real numbers using Cauchy sequences is to say that for two real numbers x and y represented by Cauchy sequences (xn) and (yn), x <= y if and only if for any positive real number there exists a positive integer N such that for all natural numbers n greater than N, xn - yn <= ε. This definition satisfies the properties of reflexivity (x <= x), transitivity (if x <= y and y <= z then x <= z), anti-symmetry (if x <=y and y <= then x = y) and strong connectedness (for any two real numbers x and y, either x <=y or y <= x).' Here is Bard: 'Let (x n) and (y n) be Cauchy sequences of real numbers. We define x n ≤ y n if and only if there exists a constant C such that ∣x n − y n∣ < C for all sufficiently large n. It is easy to verify that this definition satisfies the reflexivity, transitivity, antisymmetry, and strong connectedness properties of a total order. Here is a more intuitive explanation of this definition. If x n ≤ y n, then the sequence (x n) is "eventually dominated" by the sequence (y n). In other words, there comes a point after which the terms of (x n) are always smaller than the corresponding terms of (y n).' Second draft: 'Let (x n) and (y n) be two Cauchy sequences of real numbers. We say that (x n)≤(y n) if and only if for all ε>0, there exists an N such that for all n>N, we have ∣x n − y n∣<ε.' Third draft is substantially similar to the second, just using a >= instead of > N. So so far this seems worse than GPT.

mannerheim 4y ago
Bing follow-up: 'Can you prove those properties? 'Sure! Here's a proof for each property: Reflexivity: For any real number x represented by a Cauchy sequence (xn), we have that for any positive real number there exists a positive integer N such that for all natural numbers n greater than N, xn - xn = 0 <= ε. Thus, x <= x. Transitivity: Suppose x <= y and y <= z for real numbers x,y,z represented by Cauchy sequences (xn), (yn), and (zn) respectively. Then for any positive real number there exist positive integers N1 and N2 such that for all natural numbers n greater than N1 and m greater than N2, xn - yn <= ε/2 and ym - zm <= ε/2. Let N max(N1,N2). Then for all natural numbers n greater than N, we have xn - zn = (xn - yn) + (yn - zn) <= ε/2 + ε/2 = Thus, x <= z. Anti-symmetry: Suppose x <= y and y <= x for real numbers x,y represented by Cauchy sequences (xn) and (yn) respectively. Then for any positive real number ε, there exist positive integers N1 and N2 such that for all natural numbers n greater than N1 and m greater than N2, xn - yn <= ε/2 and ym - xm <= ε/2. Let N = max(N1,N2). Then for all natural numbers n greater than N, we have |xn - yn| = |I(xn - yn) + = I(xn - yn) + (ym - xm)| <= |xn-yn| + lym-xml<=ε/2+ε/2=ε. Since this holds true for any positive real number it follows that the sequence (|xn-yn|) converges to 0 which implies that the two Cauchy sequences represent the same real number i.e., x=y. Strong connectedness: For any two real numbers represented by Cauchy sequences (xn) and (yn), either there exists a positive integer M such that xn- yn<=0 or there exists a positive integer M such that yn-xn<=0. In the first case we have that x<=y while in the second case we have y<=x. Hope this helps!' Bard second try: Prompt: 'A total order requires a <= relation which must be reflexive, transitive, anti-symmetric, and strongly connected. On Cauchy sequences, we define a sequence {x_n} <= {y_n} if for any ε > 0, there exists N such that if n > N, then x_n - y_n <= ε. Prove that this definition of <= on Cauchy sequences is a total order.' Bard: 'Sure. Here is a proof that the definition of <= on Cauchy sequences is a total order: Reflexivity: Let x n be a Cauchy sequence. Then for any ε>0, there exists N such that if n>N, then x n − x n < ε. Therefore, x n <= x n. Transitivity: Let x n and y n be Cauchy sequences. Then for any ε>0, there exists N such that if n>N, then x n − y n < ε and y n − z n < ε. Therefore, x n − z n < 2ε. Anti-symmetry: Let x n and y n be Cauchy sequences. If x n <= y n and y n <= x n, then for any ε>0, there exists N such that if n>N, then x n − y n < ε and y n − x n < ε. Therefore, x n = y n. Strong connectivity: Let x n and y n be Cauchy sequences. Then for any ε>0, there exists N such that if n>N, then x n − y n<ε or y n − x n < ε. Therefore, x n and y n are connected. Therefore, the definition of <= on Cauchy sequences is a total order.' Reflexivity and transitivity were right for both, Bing was dead wrong about anti-symmetry, Bard gave enough to prove anti-symmetry for a reasonable definition of equality of Cauchy sequences, and neither got a proof of strong connectedness right, they essentially assert it without proving it. But that is substantially more difficult than the other properties to prove.

mannerheim 4y ago

Bing follow-up: 'Can you prove those properties? 'Sure! Here's a proof for each property: Reflexivity: For any real number x represented by a Cauchy sequence (xn), we have that for any positive real number there exists a positive integer N such that for all natural numbers n greater than N, xn - xn = 0 <= ε. Thus, x <= x. Transitivity: Suppose x <= y and y <= z for real numbers x,y,z represented by Cauchy sequences (xn), (yn), and (zn) respectively. Then for any positive real number there exist positive integers N1 and N2 such that for all natural numbers n greater than N1 and m greater than N2, xn - yn <= ε/2 and ym - zm <= ε/2. Let N max(N1,N2). Then for all natural numbers n greater than N, we have xn - zn = (xn - yn) + (yn - zn) <= ε/2 + ε/2 = Thus, x <= z. Anti-symmetry: Suppose x <= y and y <= x for real numbers x,y represented by Cauchy sequences (xn) and (yn) respectively. Then for any positive real number ε, there exist positive integers N1 and N2 such that for all natural numbers n greater than N1 and m greater than N2, xn - yn <= ε/2 and ym - xm <= ε/2. Let N = max(N1,N2). Then for all natural numbers n greater than N, we have |xn - yn| = |I(xn - yn) + = I(xn - yn) + (ym - xm)| <= |xn-yn| + lym-xml<=ε/2+ε/2=ε. Since this holds true for any positive real number it follows that the sequence (|xn-yn|) converges to 0 which implies that the two Cauchy sequences represent the same real number i.e., x=y. Strong connectedness: For any two real numbers represented by Cauchy sequences (xn) and (yn), either there exists a positive integer M such that xn- yn<=0 or there exists a positive integer M such that yn-xn<=0. In the first case we have that x<=y while in the second case we have y<=x. Hope this helps!' Bard second try: Prompt: 'A total order requires a <= relation which must be reflexive, transitive, anti-symmetric, and strongly connected. On Cauchy sequences, we define a sequence {x_n} <= {y_n} if for any ε > 0, there exists N such that if n > N, then x_n - y_n <= ε. Prove that this definition of <= on Cauchy sequences is a total order.' Bard: 'Sure. Here is a proof that the definition of <= on Cauchy sequences is a total order: Reflexivity: Let x n be a Cauchy sequence. Then for any ε>0, there exists N such that if n>N, then x n − x n < ε. Therefore, x n <= x n. Transitivity: Let x n and y n be Cauchy sequences. Then for any ε>0, there exists N such that if n>N, then x n − y n < ε and y n − z n < ε. Therefore, x n − z n < 2ε. Anti-symmetry: Let x n and y n be Cauchy sequences. If x n <= y n and y n <= x n, then for any ε>0, there exists N such that if n>N, then x n − y n < ε and y n − x n < ε. Therefore, x n = y n. Strong connectivity: Let x n and y n be Cauchy sequences. Then for any ε>0, there exists N such that if n>N, then x n − y n<ε or y n − x n < ε. Therefore, x n and y n are connected. Therefore, the definition of <= on Cauchy sequences is a total order.' Reflexivity and transitivity were right for both, Bing was dead wrong about anti-symmetry, Bard gave enough to prove anti-symmetry for a reasonable definition of equality of Cauchy sequences, and neither got a proof of strong connectedness right, they essentially assert it without proving it. But that is substantially more difficult than the other properties to prove.