Mathematical Reasoning Class 11 contains some basic ideas of Mathematical Reasoning. In this chapter, you will learn the process of reasoning especially in the context of mathematics. In mathematical language, there are two kinds of reasoning – inductive and deductive. You might have already learned inductive reasoning in the context of mathematical induction. In this Chapter 14, you will learn some fundamentals of deductive reasoning.
Notes of Class 11 Chapter 14 Mathematical Reasoning includes identifying the mathematically acceptable statement, how to write new statements from old with respect to the negation of a statement, compound statement. Besides, you will learn how to combine the statements using some special words or phrases. In this chapter, students will learn how to validate the statements using various techniques.
Click here to get the Notes for all chapters of Class 11 Maths.
Mathematical Reasoning Class 11 Notes
Statements in Mathematics
A sentence is called a mathematically acceptable statement if it is either true or false but not both. Whenever we mention a statement here, it is a “mathematically acceptable” statement.
While studying mathematics, we come across many such sentences. Some examples are given as:
Three plus three equals six.
The sum of two positive numbers is positive.
All prime numbers are odd numbers.
Of these sentences, the first two are true, and the third one is false. There is no ambiguity about these sentences. Therefore, they are statements. Also, when a sentence is ambiguous, such a sentence is not acceptable as a statement in mathematics.
New Statements from Old
In this section, students can learn the method for producing new statements from those that we already have.
Negation of a statement
The denial of a statement is called the negation of the statement. In other words, if p is a statement, then the negation of p is also a statement and is denoted by ∼ p, and read as ‘not p’.
While forming the negation of a statement, phrases like, “It is not the case” or “It is false that” are also used.
Let us consider the statement:
p: Bangalore is a city.
The negation of this statement is
~p: It is not the case that Bangalore is a city
This can also be written as
~p: It is false that Bangalore is a city.
This can simply be expressed as
~p: Bangalore is not a city
These are the different ways to write the negation of a given sentence.
Many mathematical statements are obtained by combining one or more statements using some connecting words like “and”, “or”, etc.
Now, suppose two statements are given as below:
p: 5 is an odd number.
q: 5 is a prime number.
We can combine these two statements with “and”.
r: 5 is both an odd and prime number.
This is a compound statement.
From this, we can write the following definition:
A Compound Statement is a statement that is made up of two or more statements. In this case, each statement is called a component statement.
In the above example, p and q are component statements, whereas r is a compound statement.
We can observe some of the connecting words in compound statements like “And”, “Or”, etc. These are called connectives.
Whenever we write compound statements, it is necessary to recognise the role of these connecting words.
The word “And”
Let us look at a compound statement with “And”.
p: 30 is a multiple of 5, 6 and 7.
This statement has the following component statements
q: 30 is a multiple of 5.
r: 30 is a multiple of 6.
s: 30 is a multiple of 7.
Here, we know that the first is false while the other two are true.
Rules of “And”
We have the following rules about the connective word “And”
- The compound statement with ‘And’ is true if all its component statements are true.
- The component statement with ‘And’ is false if any of its component statements are false (this includes the case that some of its component statements are false or all of its component statements are false).
The word “Or”
Compound statements can also be written using the word “Or”. However, in this case, we can determine whether an inclusive “Or” or exclusive “Or” is used.
Let us consider the following examples to understand the difference between inclusive and exclusive Or.
Example 1: The school is closed if it is a holiday or a Sunday.
Here also “Or” is inclusive since school is closed on holidays as well as on Sunday.
Example 2: Two lines intersect at a point or are parallel.
Here “Or” is exclusive because two lines cannot intersect and parallel together.
It is essential to note the difference between these two ways because we require this when we check whether the statement is true.
Rules for the compound statement with ‘Or’
- A compound statement with an ‘Or’ is true when one component statement is true or both are true.
- A compound statement with an ‘Or’ is false when both the component statements are false.
Quantifiers are phrases like “There exists” and “For all”. A word closely connected with “there exists” is “for every” (or for all). Hence, the words “And” and “Or” are called connectives and “There exists” and “For all” are called quantifiers.
Click here to understand the connectives and compound statements in mathematics.
Implications in Mathematical Reasoning
It is possible to write statements with implications such as “If ”, “only if ”, “ if and only if ”. The statements with “if-then” are very common in mathematics.
A sentence with if p, then q can be written in the following ways.
- p implies q (denoted by p ⇒ q)
- p is a sufficient condition for q
- q is a necessary condition for p
- p only if q
- ∼q implies ∼p
‘If and only if’, represented by the symbol ‘⇔‘ means the following equivalent forms for the given statements p and q.
(i) p if and only if q
(ii) q if and only if p
(iii) p is necessary and sufficient condition for q and vice-versa
(iv) p ⇔ q
Contrapositive and converse
Contrapositive and converse are certain other statements which can be formed from a given statement with “if-then”.
- The contrapositive of a statement p ⇒ q is the statement ∼ q ⇒ ∼p .
- The converse of a statement p ⇒ q is the statement q ⇒ p.
- p ⇒ q together with its converse, gives p if and only if q
Click here to learn about contrapositive and converse statements in detail.
The following methods are used to check the validity of statements:
(i) direct method
(ii) contrapositive method
(iii) method of contradiction
(iv) using a counter example.
Learn more about Validating statements in Mathematical reasoning in detail here.
Mathematical Reasoning Questions for Class 11
Go through the practice questions given below:
- Show that the statement “For any real numbers a and b, a2 = b2 implies that a = b” is not true by giving a counter-example.
- Rewrite the following statement with “if-then” in five different ways conveying the same meaning.
- Check whether the following pair of statements are negations of each other. Give reasons for your answer.
(i) x + y = y + x is true for every real number x and y.
(ii) There exists real numbers x and y for which x + y = y + x.
- Find the component statements of the following compound statements and check whether they are true or false.
(i) Number 3 is prime or it is odd.
(ii) All integers are positive or negative.
(iii) 100 is divisible by 3, 11 and 5.
If a natural number is odd, then its square is also odd