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1. Which of the following statements are true and which are false? Give reasons for your
answer. (20)
i) ‘x
2 +y
2 −3 is not dividible by 4.’ is mathematical statement.
ii) The number of onto functions from {1,2,3,4,5,6} to {a,b, c,d} is 4!S
4
6
.
iii) The generating function associated with a sequence can never be a polynomial.
iv) K4,4 is non-planar.
v) Every bipartite graph with odd number of vertices is non-hamiltonian.
vi) an = an
2
+n,a1 = 0, where n is a power of 2, is a linear recurrence relation.
vii) The generating function of the sequence {1,2,3,4,...,n...} is (1−z)
−2
.
viii) If g(x) is the generating function for {an}n≥1, then (1−x)g(x) is the generating
function for the sequence {bn}n≥1 where bn = an −1,∀n.
ix) If a graph is isomorphic to its complement, then it has odd number of vertices.
x) Every 3-colourable graph is 4-colourable.
Suppose that Q(x) is “x+1=2x”, where x is a real number. Find the truth value of the following statement:
a) Q(2) b) ∀𝑄(𝑥) c) ∃𝑄(𝑥)
Prove the equivalence of the following in three different ways (truth table, simplification,
each is a logical consequence of the other): p → (q ∨ r) ≡ (p ∧ ~q) → r.
Let R be the relation on the set A = {1,2,3,4,5,6,7} defined by the rule (a,b) equivalent to R, if the integer (a-b) is divisible by 4,
List the elements of R and its inverse
- Let p and q be the propositions
p : I bought a lottery ticket this week.
q : I won the million dollar jackpot.
Express each of these propositions as an English sentence.
a) ¬p
b) p ∨ q
c) p → q
d) p ∧ q
e) p ↔ q
f ) ¬p → ¬q
g) ¬p ∧ ¬q
h) ¬p ∨ (p ∧ q)
determine the truth value of each of the following truth values
if 2+3=5 then 2x3=6
Show that -p → (q + r) and q→ (p V r) are logically equivalent.
Self - Assessment
A. List the members of the following sets
1. {x| x is real numbers and x2 = 1}
2. {x| x is an integer and -4 < x ≤ 3}
B. Use set builder notation to give description of each of these sets.
1. {a, e,i ,o, u}
2. {=2, -1, 0, 1, 2}
C. Let A= (a, b, c), B = (x, y) and C = (0, 1)
Find:
1. A U C
2. C x B
3. B – A
4. (A ∩ C) U B
D. Find these terms of the sequence (An}, where An = 2(3)n + 5
1. A0
2. A5
3. A3
4. 8th term
5. 2nd term
6. Sum of the sequence
E. Given the following set:
2. X = {-1, 0, 1, 2, 3, 4, 5} defined by the rule (x, y) ∈R if x ≤ y
F. List the elements of R
G. Find the domain of R
H. Find the range of R
I. Draw the digraph
J. Properties of the Relation
Show step by step:
find the inverse of {(-3,1) (2,4) (8,9)}
Let R be the relation on the set A = {1,2,3,4,5,6,7} defined by the rule (a,b) belongs to R, if the integer is divisible by 4,
List the elements of R and its inverse
