Search & Filtering
Show that (p → q) → (r → s) and (p → r) →
(q → s) are not logically equivalent.
The dual of a compound proposition that contains only the
logical operators ∨, ∧, and ¬ is the compound proposition
obtained by replacing each ∨ by ∧, each ∧ by ∨, each T
by F, and each F by T. The dual of s is denoted by s
∗.
Given three sets A, B, and C. Suppose we know that the union of the three sets has cardinality 182.
Further, |A| = 92, |B| = 41, |C| = 118. Also, |A ∩ B| = 15, |A ∩ C| = 42, and |A ∩ B ∩ C| = 10. Find
|B ∩ C|.
Determine the truth value of each of these statements if the domain for all variables consists of all integers. a) ∀n(n2 ≥ 0) b) ∃n(n2 = 2) c) ∀n(n2 ≥ n) d) ∃n(n2 < 0)
Identify whether the given path in the graph is (a) A simple path (b) A cycle (c) A simple cycle (i) (b, b) (ii) (a, d, c, d, e) (iii) (e, d, c, b) (iv) (d, c, b, e, d) (v) (a, d, c, b, e)
20. Give an example of a function from N to N that is
a) one-to-one but not onto.
b) onto but not one-to-one.
c) both onto and one-to-one (but different from the iden-
tity function).
d) neither one-to-one nor onto.
Suppose T(n) and f(n) and two functions. Write asymptotic notations (Ο, Ω, Θ) using these two functions and explain the growth rate of these functions in each notation.
Which is the most suitable graph representation scheme for a dense graph? Draw its
representation with the help of an example. What is the space complexity of a such
graph representation scheme?
Use mathematical induction to prove the following:
An =
(1 3n − 1
0 3n)
for all integers n ≥ 1, where A =
(1 2
0 3)
After you have proved the above statement using mathematical induction, write a
MATLAB program to verify that the formula holds for n = 1, 2, . . . , num, where num is
entered by the user. That is, for each n = 1, 2, . . . , num, get MATLAB to compute A
n and then check that the result is equal to
(1 3n − 1
0 3n)
Run the program with num = 10.
Show the following operations on set(s)-
- Complement of a set
- Symmetric difference of two sets
Define the following terms using mathematical notations. Also provide example for each
term-
Subset
Universal set
Power set
