COM103: Discrete Structures for Computer Science
Problem Set 1
1. State with reasons whether the following assertions are
propositions or not.
(a) This statement is false.
(b) 1 + 1 = 3.
(c) If 1 + 1 = 3, then pigs can fly.
(d) x2 + y2 = 1.
2. If job A arrives at the printer before job B, then A will be
printed before B. Write theconverse and the contrapositive of this
statement in English.
3. Prove the following equivalences:
(a) p q q p.(b) p q (p q) (p q).(c) x(P (x) Q(x)) x(P (x)
Q(x)).
4. Find a propositional formula on three variables p, q, r such
that it is true if and only if exactlytwo of p, q, r are set to
true.
5. Show that (p q) (p r) (q r) is a tautology.
6. The compound proposition (p q) (r (s t)) is false for exactly
one assignmentof truth values to p, q, r, s, t. Find this
assignment without constructing the truth table.
7. Determine whether the following compound proposition is a
tautology, a contradiction, or acontingency: ((s p) (s p)) ((p q)
r).
8. Let Z be the universe of discourse. Which of the following
are true?(a) xy(x > y); (b) xy(x > y).
9. Every Indian has a dream. Express this proposition in logic
using the predicate H(p, d)that means p dreams of d and two
universes of discourse: the universe I of Indians and theuniverse D
of dreams.
10. Let M(x) denote x is a man in town and S(x, y) denote x
shaves y. Express the followingstatement in predicate logic: The
barber is a man in town who shaves all those, and onlythose, men in
town who do not shave themselves.
11. Give logical expressions for the following statements. Use
quantifiers, connectives, and thepredicates P (x) and H(x) which
mean x passed the class and x did all of the homework.
(a) Every student that passed the class did all of the
homework.
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(b) There was a student that passed the class, but did not do
all of the homework.
12. Let F (a, b) denote that a and b are friends. Assuming the
domain is the set of all students ina school, translate the
following into English:
xyz((F (x, y) F (x, z) (y 6= z)) F (y, z)).
13. Prove without using the resolution rule (but using other
logical equivalences and inferencerules) that q r follows from p q
and p r.
14. Write a program for checking whether a propositional formula
(on n variables) is satisfiable.Can you write a program that
essentially consists of two subroutines?
15. How many truth tables are possible on n variables? Provide
justifications.
16. Prove or disprove the claim that x(P (x) Q(x)) is logically
equivalent toxP (x) xQ(x).
17. Show logically that the conclusion Hummingbirds are small
follows from the premisesAll hummingbirds are richly colored.No
large birds live on honey.Birds that do not live on honey are dull
in color.
18. prove that if 0 x 2, then x3 + 4x2 + 1 > 0.
19. Prove that
3 is irrational.
20. Prove that log2 3 is irrational.
21. Prove that the following is true for any positive integer
n:12 22 + 32 . . . + (1)n1n2 = (1)n1 n(n+1)2 .
22. Prove by induction: For all n Z+, 13 + 23 + 33 + . . . + n3
=[
n(n+1)2
]2.
23. Prove using induction that for all n > 1
1 +14
+19
+ . . . +1n2
< 2 1n
.
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