This document discusses various types of mathematical statements including: - Simple statements that are either true or false - Compound statements formed using connectors like "and" or "or" - Contrapositive and converse statements related to "if-then" conditionals It provides examples of each type of statement and tests the truth value of some compound statements.

1. LINEAR PROGRAMMING
Topic : Mathematical Reasoning

2. Frame : 1
In mathematics a ‘statement’ is a sentence which is either
true or false but not both.
For example, two plus two equals four is a statement.
“all prime numbers are odd numbers.” Is this a statement?

3. Frame : 2
Yes.
Exclamation, order, question, sentences involving variables
such as ‘today’, ‘tomorrow’, or ‘yesterday’ etc. are not
statements.
‘tomorrow is Friday’ is not a statement.
Which one is not a statement ?
(a) How beautiful ! (b) There are 40 days in a month
(c) Fire is always hot (d) The sun is a star

4. Frame : 3
(a) How beautiful !
The denial of a statement is called the negation of the
statement. If p is a statement then ~p is denoted as the negation
of p.
For example, the negation of p : New Delhi is a city is
~p : New Delhi is not a city.
What is the negation of the statement ‘7 is rational’?

5. Frame : 4
7 is irrational.
A statement made up of 2 or more small statements is called
a compound statement. ‘and’, ‘or’ , etc. are used to form
compound statements.
For example, The sky is blue and the grass is green is a
compound statement.
Form a compound statement using the given statements :
all rational numbers are real
all real numbers are complex

6. Frame : 5
All rational numbers are real and all real numbers are
complex.
Statements which are used to form compound statements are
called component statements. The component statements of
‘the sky is blue and the grass is green’ are
The sky is blue
The grass is green
What are the component statements of ‘a square is a
quadrilateral and its 4 sides are equal.’

7. Frame : 6
The component statements are ‘a square is a quadrilateral’ and
‘4 sides of a square are equal.’
The compound statement with ‘and’ is true if all its component
statements are true. The compound statement with ‘and’ is
false if any of its component statement is false.
Check whether the compound statement is true or false
“0 is less than every positive integer and every negative
integer.”

8. Frame : 7
False
A compound statement with an ‘or’ is true when one
component statement is true or both the component
statements are true. A compound statement with ‘or’ is false
when both component statements are false.
“125 is a multiple of 7 or 8.” check whether the statement is
true or false.

9. Frame : 8
False
Consider the following ‘if-then’ statement.
‘if the physical environment changes, then the biological
environment changes.’ Then the contra positive of the
statement is ‘if the biological environment does not changes
then the physical environment does not changes.’ Both these
statements convey the same meaning.
What is the contra positive of the statement “if a number is
divisible by 9, then it is divisible by 3?”

10. Frame : 9
If a number is not divisible by 3, then it is not divisible by 9.
Consider the following ‘if-then’ statement.
‘if a number is not divisible by 3, then it is not divisible by 9.’
The converse of this is ‘if a number is not divisible by 9, then
it is not divisible by 3.’
What is the converse of the statement “if a number n² is even,
then n is even.”

11. Frame : 10
If a number n is even then n² is even.

12. Prepared By :
NASEERA P P
No : 28
B Ed Mathematics
KSTC Taliparamba.