Set Theory : Well Orderings and Transfinite Induction

Study concepts, example questions & explanations for Set Theory

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Example Questions

Example Question #1 : Well Orderings And Transfinite Induction

Determine if the following statement is true or false:

Let  and  be well ordered and order isomorphic sets. If  and  are also order isomorphic sets, then  and  are also order isomorphic.

Possible Answers:

False

True

Correct answer:

True

Explanation:

This is a theorem for well ordered sets and the proof is as follows.

First identify what is given in the statement.

1. The sets are onto order isomorphisms

 and 

2. The goal is to make an onto order isomorphism. Let us call it .

Thus,  can be defined as the function,

To show  is a well defined, one-to-one,  function on  since  and  are one-to-one, the following is performed.

                                

Thus,  is one-to-one.

Now to prove  in onto  if  

for some

thus

proving  is onto .

Lastly prove ordering.

                                             

                                             

Thus proving  is isomorphic. Therefore the statement is true.

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