Sample 2nd Midterm

# Sample 2nd Midterm - SAMPLE PROBLEMS FOR THE SECOND M114S...

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Unformatted text preview: SAMPLE PROBLEMS FOR THE SECOND M114S MIDTERM Forget about the numbers—the problems come from previous midterms and finals and I have not bothered to change their numbers Problem 3 . Consider the following binary relation on the set ( N → N ) of all functions on the natural numbers: f ∼ g ⇐⇒ { n | f ( n ) = 0 } = c { m | g ( m ) = 0 } . 3a . Prove that ∼ is an equivalence relation on ( N → N ). 3b . Prove that ∼ has a countable quotient, i.e., the following: there is a countable set Q and a surjection π : ( N → N ) →→ Q, such that f ∼ g ⇐⇒ π ( f ) = π ( g ) ( f,g : N → N ) . Problem 4 . Compute the following cardinal numbers, where ℵ = | N | , c = |P ( N ) | . Each step in the computation must be justified by some theorem we have proved, but it is not necessary to quote and state all these theorems—just don’t put down equations which cannot be justified easily from known results. All the answers are among the following cardinal numbers: ℵ , c , 2...
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Sample 2nd Midterm - SAMPLE PROBLEMS FOR THE SECOND M114S...

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