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Math 347H Assignment 1
1. Do exercise 1.3.9. (i), (ii), (v), (x), (xii)
2. Suppose α, β are irrational numbers. Can α β be rational?
3. Show that R = {(x, y) ∈ R × R : x ≤ y} is a reflexive and transitive relation on R. Is it symmetric? Why or why not? Is it an equivalence relation?
4. Use inclusion-exclusion and the fundamental counting principle (Theorem 1.4.7) to determine the number of rearrangements of M,A,T,H,I,S,F,U,N where none of “MATH”, “IS” or “FUN” appear as a substring.
5. Prove that if R, S ⊆ X × X are equivalence relations on X then R ∩ S is an equivalence relation. Is R ∪ S an equivalence relation? If yes, give a proof. If no, give a counterexample.
6. For each n ≥ 1, count the number of pairs (A, B) such that ∅ ⊆ A ⊆ B ⊆ {1, 2, . . . ,