This document contains a summary of 5 tutorials on discrete mathematics concepts: 1. A Venn diagram problem involving sets of people who like different drinks is presented along with problems involving sets of boys playing different games. 2. Relations and equivalence relations are discussed along with examples testing whether certain relations are equivalence relations. 3. Mathematical induction and logic proofs involving implications, conjunctions and disjunctions are provided. 4. Divisibility proofs and logic proofs with quantifiers are given. 5. Concepts of tautology, contradiction and Hasse diagrams for partial orders are defined along with examples constructing Hasse diagrams.

1. Discrete Mathematics Tutorials
Kushal Gupta
Assistant Professor
Computer Science & Engg. Dept.
DIT University, Dehradun
Tutorial 1:
1) In a survey of 85 people it is found that 31 like to drink milk, 43 like coffee and 39 like tea. Also 13 like
both milk and tea, 15 like milk and coffee, 20 like tea and coffee and 12 like none of the three drinks.
Find the number of people who like all the three drinks. Display the answer using a Venn diagram.
2) In a class of 60 boys, 45 boys play cards and 30 boys play carrom. How many boys play both games?
How many play cards only and how many play carrom only?
3) If A and B are two subsets of a universal set then prove that A-B = A∩ B'
Tutorial 2:
4) Let X be the set of all programs of a given programming language. Let R the relation on X be defined
as P1 R P2 if P1 and P2 give the same output on all the inputs for which they terminate. Is R an
equivalence relation? If not which property fails?
5) Let m be a positive integer with m>1. Determine whether or not the following relation is an
equivalent relation.
R = {(a, b) a ≡ b(mod m)}
6) Let A = {a,b,c,d,e} and R={(a,a), (a,b), (b,c), (c,e), (c,d), (d,e)}
Compute (i) R2
and (ii) R∞

2. Tutorial 3:
7) Show that n3
+2.n is divisible by 3 using Mathematical Induction.
8) Show that S ∨ R is tautologically implied by (P ∨ Q)∧ (P → R) ∧ (Q → S)
9) Show that
⌐ ( P ∧ Q) → (⌐ P ∨ (⌐P ∨ Q ) ⇔ (⌐P ∨ Q)
Tutorial 4:
10) Show that xn
-yn
is divisible by (x - y) for all positive integral values of n.
11) Show that (∃x)(P(x)∧ Q(x))⇒ (∃x) P (x)∧ (∃x)Q (x)
12) Define tautology and contradiction. Show that “If the sky is cloudy then it will rain and it will not
rain”, is not a contradiction.

3. Tutorial 5:
13) Draw the Hasse diagram for the poset
14) Consider the set A = {2, 7, 14, 28, 56, 84}
Hasse diagram for the poset (A,
Tutorial 6:
15) Consider the following Hasse di
(i) all upper bounds of B;
(ii) all lower bounds of B;
(iii) the least upper bound of B;
(iv) the greatest lower bound of B;
16) Which of the partially ordered sets in figures (i), (ii)
13) Draw the Hasse diagram for the poset (℘(A),⊂) where A = {1, 2, 3, 4}and ℘(A) is the
A = {2, 7, 14, 28, 56, 84} and the relation a ≤ b if and only if a divides b. Give the
(A, ≤ )
15) Consider the following Hasse diagram Fig. and the set B= {3, 4, 6}. Find, if they exist,
(iii) the least upper bound of B;
(iv) the greatest lower bound of B;
hich of the partially ordered sets in figures (i), (ii) and (iii) are lattices? Justify your answer
is the power set of A.
if and only if a divides b. Give the
agram Fig. and the set B= {3, 4, 6}. Find, if they exist,
and (iii) are lattices? Justify your answer