Module 1 Test paper
Time:3hours
Total
: 100 marks
Section A
1.
Write the converse of the following implication
“ when I stay up late it is necessary that I sleep until noon”
2.
Translate the following statement into English
if the universe of discourse for each variable consists of all real numbers Ɐxꓱy(x<y)
3.
What is meant by tautology?
4.
For the propositions p and q write p v ┐q
as an English sentence
P: I own a car
Q: I own a dog
5.
Write the logically equivalent expression for (┐p
v q)→r
6.
State distributive laws in set theory
7.
If the truth value of ( p ^ q)→(r
v s) is false then what will be the truth values of its components
8.
Write the inverse of the statement “if there is
forests then there is rain”
9.
Write the equivalent expression of ┐(p^q)
using De morgan’s law
10.
When you say that an argument is a fallacy?
11.
What is a statement?
Section B(2*20=40)
12.
What is meant by Boolean expression? Give an
example.
13.
Draw the truth tables of conditional and
biconditional statements.
14.
Write an equivalent formula for p v (s↔t)
which does not involve biconditional connective and conditional connective
15.
Express the Boolean expression E(x,y,z) = (x’
+y)’ + x ‘ y in complete sum of products form
16.
Let p and q be the propositions
P: it is below freezing
Q: it is snowing
Write the following propostions using P and
Q and logical connectives
a.
It is below freezing and snowing
b.
It is below freezing but snowing
17.
Symbolize and negate the propositions
a.
All boys can run faster than all girls
b.
Some girls are more intelligent than all boys
18.
Express the Boolean expression E(x,y,z) = x(y’
z)’ in complete sum of products form
19.
Find the contrapositive of the following
statements:
a.
If jack is a poet then he is poor
b.
If x is less than zero then x is not positive
20.
Show that p↔q ≡ (p v q) →
(p ^q)
21.
Obtain disjunctive normal forms of
a.
P ^ (P → Q)
b.
┐(P v Q) ↔ (P ^ Q)
Section C(7*3 = 21)
22.
Explain the resolution principle with a suitable
example
23.
Prove by induction that n3 < 3n
for n≥4.
24.
Prove by contradiction “ if 3n + 2 is odd then n
is odd”
25.
Prove that 1 + 3 + 5+ … + 2n-1 = n2
26.
Test the validity of the argument
P→Q
Q
_____
P
27.
Show that the propositions ┐(P^Q)
and ┐P
v ┐Q
are logically equivalent
28.
Verify the proposition p v (p^q) is a tautology.
Section D
29.
Prove that 
is irrational by giving a proof by
contradiction.
is irrational by giving a proof by
contradiction.
30.
Verify that the propostion ( p^q) ^ ┐(p
v q) is a contradiction.
31.
Differentiate the contrapositive and
contradiction method of proof
32.
Check the validity “ All the living
beings are either an animal or a plant. Johns gold fiah is alive but not a
plant . all animals are cruel. Therefore there are cruel beings in the world.”
33.
Test the validity of the argument
If 6 is even then 2 does not divide 7.
Either 5 Is not prime or 2 divides 7. But 5 is prime, therefore 6 is odd.
34.
Test the validity of the argument
If I study then I will not fail in
mathematics
If I do not play basketball then I study
But I failed in mathematics
Therefore I played basketball.
35.
Test the validity of the argument
If a man is a bachelor, he is unhappy
If a man is unhappy, he dies young
_______________________________
Bachelors die young
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