__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:**

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

^{3}= 64

**Example 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: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:Solution:

**(a)**11 is an old number or a prime number.

**(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

**= statement 1 and***p**= statement 2.***q**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 true15 – 7 = 8 ← (

**)***q*is trueTherefore 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 falseTherefore 5 > 3 and –4 < –5 is a

**false statement**. (‘**)***p*and*q*’ is false

(c)

(c)

Hexagons have 5 sides. ← (

**)***p*is falseEach of the interior angles of Hexagon is 90

^{o}. ← (**)***q*is falseTherefore 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

**= statement 1 and***p***= statement 2.***q*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 true60 is divisible by 9 ← (

**)***q*is falseTherefore, 60 is divisible by 4 or 9 is a

**true statement**. (‘**)***p*or*q*’ is true

**(b)**

5

^{3}= 25 ← (**)***p*is false4

^{3}= 64 ← (**)***q*is trueTherefore, 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 falseTherefore, 5 + 7 > 14 or √9 = 2 is a

**false statement**. (‘**)***p*or*q*’ is false