1) Decide which of the following are propositions (Y/N) and give the truth value:

a) 127 is an even integer ___ b) All triangles have 4 sides or more___

2) Negate the following statements:

a) John is smart or Fred is not tall ...............................................................................................

b) Some animals are intelligent .....................................................................................................


3) Is ((not P) implies Q) equivalent to ( P implies ( not Q)) ? ................

4) List all elements of set S, where S = { x | where x belongs to N, -3 =< x < 5 }

S = {.........................................................................}

5) Fill in the blanks: U: union, {}: empty set, P: power set, -: set difference, x: Cartesian product.

a) A U A = _____ b) A intersect {} = _____ c) A U {} = _____

d) P( P ({})) = _____ e) P({a,{b}}) = _____

f) {a,{b}} - {a,{b},c} = _____ g) {1,a} x {a} = _____

6) Given the proposition P(n) " If n3 (n cubed) is odd then n is odd "

a) Prove P(n) by contraposition:

b) Prove P(n) contradiction:

7) Prove by induction: 1² + 2² + 3²... +n² = n(n+1)(2n+1)/6, for all n >= 1. 

8) Given the English language alphabet. Is the set of all possible strings of letters (even the ones that infinitely long) countable?