Help with using the Schroeder-Bernstein Theorem? Corresponding Counts [3 points] Prove that <m

Jonathan Kent 2022-05-23 Answered
Help with using the Schroeder-Bernstein Theorem?
Corresponding Counts [3 points]
Prove that | { x R | 0 x 1 } | = | { x R | 4 < x < 7 } | .
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Answers (2)

Terrance Phillips
Answered 2022-05-24 Author has 10 answers
Step 1
The injection x x + 5 shows that we have an injection from [0, 1] into (4, 7), so by definition | [ 0 , 1 ] | ( 4 , 7 ) | .
Step 2
The injection x x 7 shows that we have an injection from (4, 7) into [0, 1] (the image of (4, 7) is ( 4 7 , 1 ) [ 0 , 1 ], so | ( 4 , 7 ) | | [ 0 , 1 ] | .
Now Cantor-Bernstein does the rest.
It's a handy tool to not have to give an exact bijection between these two sets.
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Andy Erickson
Answered 2022-05-25 Author has 3 answers
Step 1
The main strategy for these kinds of problems is that you want to show that one set is one-to-one and the other is one-to-one as well. You show this by finding a function the maps from one set to the other.
For this question let's say that the interval [0,1] is A, and the other (4,7) is B. Showing that | A | | B | is saying that this function is one-to-one. To show this:
[ 0 , 1 ] ( 4 , 7 ), let's find a function that will map every input from [0, 1] to (4, 7). A function that would accomplish this would be f ( x ) = x + 5. To show this you could make a simple diagram like: 0 5 .5 5.5 1 6
Step 2
For ( 4 , 7 ) [ 0 , 1 ], a potential function could be: f ( x ) = x 4 1. Again, drawing a diagram like:
5 .25
By showing [ 0 , 1 ] ( 4 , 7 ) is one-to-one, then | [ 0 , 1 ] | | ( 4 , 7 ) | and have shown ( 4 , 7 ) [ 0 , 1 ] meaning | ( 4 , 7 ) | | [ 0 , 1 ] | is also one-to-one, then we can conclude using Schröder-Bernstein that | [ 0 , 1 ] | = | ( 4 , 7 ) | .
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