Counting

7 women and 7 men are in the department of mathematics. How many ways are there to select a committee of 5 members if at least 1 woman and least 1 man must be on the committee?

Proof : Equivalence Relations and Divisibility

Prove 8|5^(n+1) +(2)3^n + 1, n Є N

Equivalence Relations and Classes

For m, n, in N define m~n if m^2 — n^2 is a multiple of 3. (a.) Show that ~ is an equivalence relation on N. (b.) List four elements in the equivalence class [0]. c) List four elements in the equivalence class [1]. (d.) Are there any more equivalence classes. Explain your answer.

Modular Arithmetic Proof

Prove that if m,n Є Z and m ≡ n(mod p) then m^n ≡ n^n(mod p).

Collection of subsets for equivalence relations

Let S be a collection of subsets of X, X = US. [S is not necessarily peicewise disjoint.] xRy if x, y Є S ⊂ S. Is R necessarily an equivalence relation? Show why or why not.