August 8, 1996
1. Using the predicate symbols given below, transcribe each of the following English sentences
into a proposition of a predicate logic. The universe is the whole world.
B(x): x is a bee.
F(x): x is a flower.
K(x): x is a beekeeper.
L(x,y): x loves y.
(a) Not everything is a bee.
(b) Some bees love all flowers.
(c) Every bee loves only flowers.
(d) Every beekeeper loves bees which love flowers.
(e) No beekeeper loves bees if they don't love flowers.
2 (a) Find the power set of { , { }, 1}.
(b) Prove that .
Hint: Use proof by contradiction.
3. Prove by induction that
for all natural number n.
4 (a) Find the transitive closure of the binary relation { }.
You may use a digraph to represent your answer, if you prefer that.
(b) Determine for each relation given below whether it is reflexive, irreflexive, symmetric,
antisymmetric and/or transitive. Also which ones are a partial order and which ones are
an equivalence relation ?
(1) For any subsets A and B of a set S, iff .
(2) For any integers x and y, iff (mod 5).
(3) For any numbers x and y, iff .
5 (a) Define the set of non-negative integer multiples of 5 inductively.
(b) Prove by induction that any arbitrary non-negative integer multiple of 5 ends in
0 or 5, that is its least significant digit is 0 or 5.