HAWASSA UNIVERSITY
Collage of Natural and Computational Science
Department of Mathematics
Work sheet -1
Basic Mathematics for Natural Sciences
1. Show that the following propositions are equivalent
a) (
)
2. For statements
4.
5.
6.

(
b)
)
, and , show that the following compound statements are tautology.
(
) (
))
. b) ((
(
).
Let and be statements. Which of the following implies that
is false?
a. 
 is false.
c.
is true.
b. 
is true.
d.
 is true.
Suppose that the statements
and
are assigned the truth values
respectively. Find the truth value of each of the following statements

(
)
(a)
(
)
(c)

(
)
(b)
(
).
(d) (
)
a) If the value of
is , then what is the truth value of(
)
(
)?
b) If the value of
is , then what is the truth value of
c) If the value of
is , then what is the truth value of
Find the truth value of following compound statements using the given information.
a) (
)
, where
.
c)
(
), where
.
)
)
) (
(
 ), where
.
d) (
(
 ), where
)
) (
(
), where
.
)(
3.

(2014EC)
7. Let
(
)
))
where the domain of
is
*
+ and the domain of
determine the truth value of the following quantified statement.
)(
) (
)(
) ( ). c) (
a) (
). b) (
8. Determine the truth value of the following statements.
)(
)(
)(
a) (
).
e) (
)(
)(
)(
b) (
).
f) (
*
is
)(
and
,
.
+. Then
) (
)
).
)
)(√
)(
)(
c) (
).
g) (
)
)(
)(
)(
d) (
).
)(
)
9. Investigate the validity of the following argument forms using truth table method.
)

.
)

 .
c)



 .
10. Investigate the validity of the following argument forms using formal proof (rule of inference)
a) 
b)


.
 .
c) 
 (
d) 

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)

 .
. e)
Mathematics Dept.



 .
11. For the following arguments
i.
Identify the premises.
ii.
Write argument forms.
iii.
Check the validity using formal proof.
a) If prices are high, then wages are high. Prices are high or there are price controls.
Further, if there are price controls, then there is not inflation. There is inflation.
Therefore, wages are high.
b) If the team is late, then it cannot play the game. If the referee is here, then the team is can
play the game. The team is late. Therefore, the referee is not here.
*
+ +
12. Find the truth value of the following , if A= *
a.
( )
d)
( )
g) Set A has 15 proper subsets.
b. *
+
e) *
+
( )
c.
( )
f) n(P(A))= 32
13. How many elements does A contain if it has:
a) 64 subsets?
c) 31 proper subsets?
b) No proper subset? d) 255 proper subsets?
14. Let
{
or
}. Find P(A) and the no. of subset of A
15. If
,
*
+ and
*
+, find A and .
16. Let
*
+
{
is a positive prime factor of 66},
{
is
composite number} and
*
–
(
)
i.
d) (
)
ii.
(
)
e) A’ B’
iii.
17. Let
*
+ and
a.
* +, then
b.
, then
c.
*
+, then
18. If
, what is
?
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+. Then find each of the following.
*
+.
Mathematics Dept.