DISCRETE MATHEMATICS - SAMPLE EXAM QUESTIONS
1. Using truth tables decide whether each of the following is a tautology:
(a) (p→q)↔(q→p),
(b) (p→q)↔(p∧q),
(c) (p∨q)↔(p∧q),
(d) ((p∨q)∧r)↔((p∧r)∨(q∧r)).
2. For each of the following sentences, negate the sentence, and say whether the sentence or
its negation is true:
(a) ∀x∈Z,∀y∈Z,(x>y)←(x6=y),
(b) ∀x∈Z,∀y∈Z,∃z∈Z, z2=x2+y2.
3. Let A={1,2,3}. For each of the following relations decide if it is symmetric, transitive
or reflexive
(a) R={(1,1),(2,2),(2,1),(1,2)},
(b) R={(1,1),(2,2),(3,3),(1,2)}.
4. For each of the following relations Ron Z, determine whether the relation is reflexive,
symmetric or transitive, and specify the equivalence classes if Ris an equivalence relation
on Z:
(a) (a, b)∈ R if adivides b,
(b) (a, b)∈ R if a2=b2.
5. Let A={1,2,4,5,7,11,13}. Define a relation Ron Aby writing (x, y)∈ R if and only if
x−yis a multiple of 3.
(a) Show that Ris an equivalence relation on A.
(b) How many equivalence classes of Rare there?
6. Find whether the following yield functions from Nto N, and if so, whether they are one-
to-one, onto or both. Find also the inverse function if the function is one-to-one and onto:
(a) y=x2,
(b) y=
x+ 1,if xis odd,
x−1,if xis even,
.
7. Suppose that the set Acontains 5 elements and the set Bcontains 2 elements.
(a) How many different functions f:A←Bcan one define?
(b) How many of the functions in part (a) are not onto?
(c) How many of the functions in part (a) are not one-to-one?
8. Suppose that the set Acontains 2 elements and the set Bcontains 5 elements.
(a) How many of the functions f:A←Bare not onto?
(b) How many of the functions f:A←Bare not one-to-one?
9. Suppose that a∈Nand b∈Z. Proof that there exist unique q, r ∈Zsuch that b=aq +r
and 0 ¬r < a.
