INDIAN INSTITUTE OF INFORMATION TECHNOLOGY KOTTAYAM
IMA111 Discrete Mathematics
Odd Semester 2023-24: Assignment 1
1. Find a disjunctive normal form of
p → ((p → q) ∧ ¬(¬q ∨ ¬p))
2. Find a conjunctive normal form of
(p ∨ ¬(q ∨ r)) ∨ (((p ∧ q) ∨ ¬r) ∧ p)
3. Conclude S → r from the hypotheses
p → (q → r), p ∨ ¬s and q
4. Analyse whether the following argument is valid or not.
If the entry is small or giant, then the output is predictable. The output is not negative. If the output is predictable,
then it must be negative. Therefore, the entry is not small.
5. Prove by mathematical induction
7
1 5 1 3
n + n + n
5
3
15
is a natural number.
6. Show that
1
1
1
n
+
+ ··· +
=
1.2 2.3
n(n + 1)
n+1
by mathematical induction.
7. A survey of 550 television watchers produced the following information: 285 watch football games, 195 watch hockey
games, 115 watch baseball games, 45 watch football and baseball games, 70 watch football and hockey games, 50 watch
hockey and baseball games, 100 do not watch any of the three games.
ˆ How many people in the survey watch all three games?
ˆ How many people watch exactly one of the three games?
8. Let {X = 1, 2 · · · 7} and R = {(x, y) : x − yisdivisibleby3}. Show that R is an equivalence relation.
9. Each of the following defines a relation on positive integers N.
ˆ x is greater than y.
ˆ x+y=10.
ˆ xy is the square of an integer.
ˆ x+4y=10
Find which of the relations are reflexive, symmetric, antisymmetric and transitive.
10. Given A = {1, 2, 3, 4} and B = {x, y, z}. Let R be the following relation from A to B:
R = {(1, y), (1, z), (3, y), (4, x), (4, z)}.
Find
ˆ matrix of the relation
ˆ draw the digraph
ˆ inverse of R
ˆ domain and range of R.