Proficiency #13. [Showing Sets are Subsets] Let S = {x € Z | x = 8 mod 9} and let T = {x € Z|x = 2 mod 3}. Prove that S CT.

Let S be the set of all strings in a's and b's and let L: S- Z be the length function: For all strings s E S, L(s) = the number of characters in s. Let T: Z+ {0, 1, 2, 3, 4} be the mod 5 function: For every integer n, T(n) = n mod 5. (a) (To L)(bbbbaa) = (b) (To L)(babbbaaa) = (c) (To L)(abbab)

uestion 1 2 2 3 4 5 6 7 9 10 11 12 13 14 15 16 17 18 19 20 8 9 A preimage can never be congruent to its image. O True O False

2) Let X be a finite set. Show by induction on n = |X| that %3D |P(X)| = 2\X|. 2X1.