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Prove the following?

Prove the following?-example-1
User Ruhungry
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  • Answer:

Hence, The Isomorphism between Two-Well-Ordered Sets W1 and W2 is UNIQUE

  • Explanation:

Make a Plan: Understand the Problem

We need to Prove if Two-Well-Ordered Sets W1 and W2 are ISOMORPHIC, Then there is ONLY ONE ISOMORPHISM.

  • Solve the problem:

PROVE the Statement

1 -Assume there are Two Isomorphisms f and g between W1 and W2

2 - Define a Set:

A = f(x) ≠ g(x)

3 - If A is Empty, Then f(x) = g(x) for all x ∈ W1, and the Isomorphism is Unique.

4 - If A is Non-Empty, Then Since W1 is Well-Ordered, There Exist a Least Element a ∈ A

5 - Since f and g are Isomorphisms, f(a) and g(a) are the Least Elements in their respective images in W2.

6 - Since f(a) ≠ g(a), One of them, say f(a), is less than g(a).

7 - Let b = f^-1(g(a)). Then b > a because f is an Isomorphism and f(a) < g(a).

8 - Since b > a, We now have f(b) = g(b) because, a is the Least Element in A.

9 - However, f(b) = g(a), Which Contradicts the Fact that f and g are Isomorphisms

  • Draw the conclusion:

Hence, The Isomorphism between Two-Well-Ordered Sets W1 and W2 is UNIQUE

I hope this helps you!

User Noland
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