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Stupid questions?

Stupid questions?. Are there more integers than even integers? Are there less primes than integers? Are there more rational numbers than integers? Are there more real numbers than rational numbers? Are there more rational numbers or irrational numbers? Why?. Stupid questions?.

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Stupid questions?

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  1. Stupid questions? • Are there more integers than even integers? • Are there less primes than integers? • Are there more rational numbers than integers? • Are there more real numbers than rational numbers? • Are there more rational numbers or irrational numbers? Why?

  2. Stupid questions? • Are there more integers than even integers? This question seems stupid! But – we define cardinality of a set A, |A|, in the following way: |A| = |B| if and only if there is a one-one correspondence (1-1C) between the members of A and the members of B.

  3. Set Theory Here we see a bijection between the sets X and Y – the mapping is “one-one” and “onto”, Meaning every member of X corresponds to one and only one member of Y and vice versa. The existence of the bijection proves |X|=|Y|. Cantor thinks… What if X and Y are infinite sets – like the set of integers {1,2,3,4,5…} and the set of even integers {2,4,6,8,…} You can say that X is bigger than Y, but since the bijection exists – it doesn’t make sense! THE TWO SETS ARE THE SAME SIZE!

  4. Set Theory The cardinality of N, |N| = |2N| (if 2N is the set of even natural numbers). 2 challenges for you: • Is |Z| = |N|? • Is |Q| = |N| ? • Hints: remember that equal cardinalities means there is a 1-1C, so try to create a 1-1C. Try to prove that they are both true (even though the second one looks very unlikely)!

  5. Shocking #1: |Q|=|N|

  6. Shocking #1: |Q|=|N|

  7. Not shocking: |R|>|N| “reductio ad absurdum” = proof by contradiction The most beautiful proof in Mathematics? Cantor’s Diagonal Argument

  8. Not shocking: |R|>|N| Concept check… • Are there more rational numbers or irrational numbers? Why? • Side note: |R| is sometimes called the “cardinality of the continuum” and is written as c.

  9. “Je le vois, mais je ne le crois”:Shocking #2|R2|=|R| • This one you can try yourself! It’s not easy so here’s some hints. • Write each point on the plane as a pair of coordinates (x,y) • Think about how to match each point (x,y) with a single real number • Remember it needs to be a bijection (one-one and onto) • Use the decimal representation of real numbers, just as we did for the diagonal argument

  10. Bigger and bigger infinities! • The power set of a set A is the set of all subsets of A • What is the size/cardinality of the set of all subsets of the set {1,2,3,4}? • What is the size/cardinality of the set of all subsets of the set {1,2,3,…25}? • The “set of all subsets” of A is called the “power set” of A and is written P(A) Now for your biggest challenge… Prove |P(N)| > |N|

  11. A final mystery… It can be shown that c = |P(N)| so we have two “smallest” infinities: |N| < c < … The Continuum Hypothesis There is no set S such that |N| < |S| < c < … Cantor spent literally years trying to prove this, and failed…

  12. Links http://plus.maths.org/content/glimpse-cantors-paradise http://duartes.org/gustavo/blog/category/compsci

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