Search & Filtering
Question 1. [20 Marks]
a) Let p, q, and r be the propositions:
p = "the flag is set"
q = "I = 0"
r = "subroutine S is completed"
Translate each of the following propositions into symbols, using the letters p, q, r and logical connectives.
(i) If the flag is set, then I = 0 [2 marks]
(ii) The flag is set and I = 0 if subroutine S is not completed [2 marks]
(iii) Subroutine S is completed if and only if I = 0 and flag is set [2 marks]
b) State the converse, contrapositive, and inverse of each of these conditional statements.
(i) If it snows tonight, then I will stay at home [3 marks]
(ii) I go to the beach whenever it is a sunny summer day [3 marks]
(iii) When I stay up late, it is necessary that I sleep until noon [3 marks]
c) Explain the step-by-step procedure involved in finding the inverse of an n by n square matrix. [5 marks]
Show that the following relations are Partial order relations.
(a) R on the set of integers Z, defined by, R = {(a, b) | a ≤ b}
(b) R on the set of integers Z, defined by, R = {(a, b) | a ≥ b}
(c) R on the set of positive integers N, defined by, R = {(a, b) | a divides b} [Note: It is divisibility
relation]
(d) R on the Power set of a set S, defined by, R = {(A, B) | A is a subset of B} [Note: It is set inclusion relation]
Show that the following relations are Equivalence relations. Find out the Equivalence classes.
(a) ρ = {(a, b) | a – b is an integer} on the set of real numbers R
(b) R = {(a, b) | a = b or a = –b } on the set of integers Z
(c) Congruent modulo 4 relation on Z: R = {(a, b) | a – b is divisible by 4} on set of integers Z
(d) Congruent modulo 5 relation on Z: R = {(a, b) | a – b is divisible by 5} on set of integers Z
(e) R = {(S1, S2) | Length (S1) = Length (S2)} on the set of strings {Si} of English letters
(f) R = {(S1, S2) | If the first 3 bits of S1 and S2 are identical} on the set of all bit-strings {Si} of
length 4
(g) R = {(S1, S2) | If the first 3 bits of S1 and S2 are identical} on the set of all bit-strings {Si} of
length 3 or more
Let R = {(0, 0), (1, 1), (1, 2), (1, 3), (2, 0), (2, 2), (3, 0)} be a relation on the set {0, 1, 2, 3}. Find the
(a) Reflexive closure of R
(b) Symmetric closure of R
(c) Transitive closure of R
A relation R on a set S is called asymmetric if (a, b) is in R implies that (a, b) is not in R. Which of the
relations in Q. No. 5 is asymmetric?
(I) A relation R on a set S is called irreflexive, if no element in S is related to itself, that is if for every a
in S, (a, a) is not in R. Which of the relations in Q. No. 5 is irreflexive?
Write down the relation R on the Power-set of S = {1, 2, 3, 4}, defined by, R = {(A, B) | A and B are
subsets of S and they have the same cardinality} using any one of the methods: (i) set of ordered pairs, (ii)
directed graphs, (iii) matrix notation.
Write down the relation R on the S = {1, 2, 3, 4, 5, 6} by any one of the methods: set of ordered pairs,
directed graphs, matrix notation. R is given as follows.
(a) R = {(a, b) | a = b }
(b) R = {(a, b) | a ≠ b}
(c) R = {(a, b) | a < b}
(d) R = {(a, b) | a ≤ b}
(e) R = {(a, b) | a > b}
(f) R = {(a, b) | a ≥ b}
(g) R = {(a, b) | a = b + 1}
(h) R = {(a, b) | a + b ≤ 3}
(i) R = {(a, b) | a divides b}
Let S = {1, 2, 3, 4, 5, 6}. How many ordered pairs are there in S × S ?
What will be the composition of two functions f ° g from R to R,
(a)
f(x) = 2x + 3, g(x) = 3x + 2
(b)
f(x) = 2x + 3, g(x) = sin (x)
(c)
f(x) = sin (x), g(x) = 2x + 3
(d)
f(x) = sin (x), g(x) = x2
(e)
f(x) = |x|, g(x) = 2x + 3
