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Problem One
Theorem: For all integers a, b, c, d, and k, if a + c ≡k b + d, then a ≡k b or c ≡k d.
i. Write the negation of the theorem. Fully “distribute” the negation through the statement, as we learned in class (i.e., don’t just write “It is not the case that...” at the beginning and call it a day
ii. Write the contrapositive of the theorem.
iii. Prove the theorem, using proof by contrapositive. There are other ways to approach it, but we would like to see this style.
Problem Two
i. Let T3 = { ∅, {0}, {1}, {0, 1}, {1, 2}, {0, 1, 2} }. There are 6 distinct sets S such that T3 splits S. List them.
ii. Prove this theorem using Direct Proof approach: For all sets B and C, and all sets of sets T, if B ⊆C and T splits C, then T splits B.
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