4.3 Operations
on Statements
(A) Nagating
a Statement using ‘No’ or ‘Not’
1. Negation
of a statement refers to changing
the truth value of the statement,
that is, changing a true statement to a
false statement and vice versa, using the word ‘not’ or ‘no’.
Example 1:
Change the true value of the following statements by
using ‘no’ or ‘not’.
(a) 17 is a prime number.
(b) 39 is a multiple of 9.
Solution:
(a) 17 is not a
prime number. (True to false)
(b) 39 not is
a multiple of 9. (False to true)
2. A compound statement can be formed by combining two
given statements using the word ‘and’.
Example 2:
Example 3:
Identify two statements from each of the following
compound statements.
(a) All pentagons
have 5 sides and 5 vertices.
(b) 3^{3} = 27 and 4^{3} = 64
Solution:
(a) All pentagons have 5 sides.
All pentagons have 5 vertices.
(b) 3^{3} = 27
4^{3} = 64Example 3:
Form a compound statement from each of the following
pairs of statements using the word ‘and’.
(a) 19 is a prime number.
19 is an odd number.
(b) 15 – 5 = 10
15 × 5 = 75
Solution:
(a) 19 is a prime number and an odd number. ← (Repeated words can be eliminated when
combining two statements using ‘and’.)
(b) 15 – 5 = 10 and
15 × 5 = 75.
3. A compound
statement can also be formed by combining two given statements using the
word ‘or’.
Example 4:
Form a compound statement from each of the following
pairs of statements using the word ‘or’.
(a) 11 is an old number.
11 is a prime number.
$\begin{array}{l}\text{(b)}3=\sqrt[3]{27}\\ \text{}3=4+1\end{array}$
Solution:
(a) 11 is an old number or a prime number.
$(\text{b)}3=\sqrt[3]{27}\text{or}3=4+1$
(B) Truth
Values of Compound Statements using ‘And’
4. When two statements are combined using ‘and’, a true compound statement is obtained only if both statements are true.
5. If one or
both statements are false, then
the compound statement is false.
The truth
table:
Let p =
statement 1 and q = statement 2.
The truth values for ‘p’ and ‘q’ are as follows:
p

q

p
and q (compound statement)

True

True

True

True

False

False

False

True

False

False

False

False

Example 5:
Determine the truth value of the following
statements.
(a) 12 × (–3) = –36 and 15 – 7 = 8.
(b) 5 > 3 and –4 < –5.
(c) Hexagons have 5 sides and each of the interior
angles is 90^{o}.
Solution:
(a)
12 × (–3) = –36 ← (p is true)
15 – 7 = 8 ← (q is true)
Therefore 12 × (–3) = –36 and 15 – 7 = 8 is a true statement. (‘p and q’ is true)
(b)
5 > 3 ← (p is true)
–4 < –5 ← (q is false)
Therefore 5 > 3 and –4 < –5 is a false statement. (‘p and q’ is false)
(c)
Hexagons have 5 sides. ← (p is false)
Each of the interior angles of Hexagon is 90^{o}.
← (q
is false)
Therefore Hexagons have 5 sides and each of the
interior angles is 90^{o} is a false
statement. (‘p and q’ is false)
(C) Truth
Values of Compound Statements using ‘Or’
1. When two statements are combined using ‘or’, a false compound statement is obtained only if both statements are false.
2. f one or
both statements are true, then
the compound statement is true.
The truth
table:
Let p = statement 1 and q
= statement 2.
The truth values for ‘p’ or ‘q’ are as follows:
p

q

p or q (compound statement)

True

True

True

True

False

True

False

True

True

False

False

False

Example 6:
Determine the truth value of the following
statements.
(a) 60 is divisible by 4 or 9.
(b) 5^{3} = 25 or 4^{3} = 64.
(c) 5 + 7 > 14 or √9 = 2.
Solution:
(a)
60 is divisible by 4
← (p is true)
60 is divisible by 9 ← (q is false)
Therefore, 60 is divisible by 4 or 9 is a true statement. (‘p or q’ is true)
(b)
5^{3} = 25 ← (p is false)
4^{3} = 64
← (q is true)
Therefore, 5^{3} = 25 or 4^{3} = 64 is
a true statement. (‘p
or q’ is true)
(c)
5 + 7 > 14
← (p is false)
√9 = 2 ← (q
is false)
Therefore, 5 + 7 > 14 or √9 = 2 is a false statement. (‘p or q’ is false)