1. Last column of truth table contains all T's, hence a tautology.
2. Not valid, because one F appears in last column of truth
table.
3. False, True, True, True, True.
4. 62.
5. A = {1, 3, 5, 7, 9}, B = {2, 7, 3, 5}, C = {6, 7, 8, 9}.
6. 10x10x26x26x10x10x10x10.
7. C15,1)C(12,2)+C(15,2)C(12,1).
8. 0.0000244.
9. Pr[1 or 2] = 1/3.
10. 0.5.
11. {C(4,3)C(6,2)+C(4,4)C(6,1)}/C(10,5).
12. $0.20.
13. 3.568.
14. 0.1788.
15. 2A-B = matrix row 1: 0, -4; row 2: -6, 11.
16. AB = matrix row 1: -13, 20, 4; row 2: -2, -6, -3 (BA is
not defined).
17. matrix row S: .6, .2, .2; row O: .6, 0, .4; row R: .4,
.1, .5.
18. State 4 does not communicate with states 1, 2, and 3,
AND the matrix is not irreducible.
19. 0.36.
20. P3 = [.5, .25, .25], P = [1/3, 1/3, 1/3].
