Validating statements: Sentences that convey a specific opinion, fact, or idea are called statements. They are either true or false truth values, but not both. But how can we know when a statement is true? This is where validating statements come into play. The truth value of a statement is decided by the keywords or phrases, called logical operators, present in them. Several rules help us validate the statements depending on the type of logical operator present.

In this article, let us learn and familiarise the different rules and methods of validating statements in mathematical reasoning.

What is Validation?

As we already know, a statement can have a truth value of either true or false, but not both. Validating statements helps us recognise if a statement is true or not. They are based on the logical connectors that are present in the statement.

Example:

Statement	Truth Value
\(2\) is the smallest even and prime number.	True
If \(x\) is positive, \(x^{2}\) may be negative.	False

Note that these are all compound statements. The truth value of a compound statement depends on the truth value of its component statements.

Rules of Validation

Let us now discuss the various techniques to validate specific types of statements.

Rule 1: Statements with ‘and’

The truth table for ‘and’ is given below.

Component Statement \(1\)	Component Statement \(2\)	‘and’ Statement
T	T	T
T	F	F
F	T	F
F	F	F

Evidently, an ‘and’ statement is true if and only if both the component statements are true. Hence, to confirm a ‘ \(p\) and \(q\) ‘ statement is true, follow the steps given below.

Step 1: Show that the truth value of \(p\) is true.

Step 2: Show that the truth value of \(q\) is true.

Rule 2: Statements with ‘or’

The truth table for ‘or’ is given below.

Component Statement \(1\)	Component Statement \(2\)	‘or’ Statement
T	T	T
T	F	T
F	T	T
F	F	F

Here, observe that for the truth value of an ‘or’ statement to be true, at least one of the component statements must be true. Hence, two cases can be considered for a ‘ \(p\) or \(q’\) statement to be true.

Case 1: If \(p\) is false, then \(q\) must be true.

Case 2: If \(q\) is false, then \(p\) must be true.

Rule 3: Implication Statements

Implication statements have the logical connector ‘if-then’. There are two methods to verify if \(p\), then \(q’\). Confirming any one of these methods will prove the statement.

1. Contrapositive Method

In this method, the second component statement is assumed to be false. This is then used to prove that the first component statement is also false. When both the component statements are false, the statement holds. This method is called the contrapositive method of validation.

2. Direct Method

Here, the first component statement is assumed true. Then, the second component statement is validated based on that assumption. This method is called the direct method of validation.

Rule 4: Bi-implications

A bi-implication statement has the logical connector ‘if and only if’. In order to show that a ‘\(p\) if and only if \(q\)’ statement is true, we need to prove the following:

1. If \(p\) is true, then \(q\) is true.
2. If \(q\) is true, then \(p\) is true.

Rule 5: By Contradiction

Some mathematical statements cannot be proved directly. Hence, we prove this by contradiction. Contradiction is when you go against or opposite to the expected. To prove a statement is true by the method of contradiction, follow the given steps.

Step 1: Assume the truth value of \(p\) is not true. In other words, \(\sim p\) is true.

Step 2: Arrive at a result that contradicts the assumption.

Step 3: Conclude the contrary of the assumption, i.e. \(p\) is true.

Rule 6: By Using a Counter-Example

This method is used to show that a given statement is false. A counter-example is an example where the statement is not valid. In this method, we begin with a counter-example of the given statement, and then go on to verify the validity of the statement.

Note that, although counter-examples can be used to disprove the given statement, examples of a statement do not validate it.

Solved Examples

Q.1. Check the validity of the statement. \(p:60\) is a multiple of \(3\) and \(5\).
Sol:
Given:
\(p: 60\) is a multiple of \(3\) and \(5\)
Here, the logical connector is ‘and’
Let,
\(q: 60\) is a multiple of \(3\)
\(r: 60\) is a multiple of \(5\)
\(p: q \wedge r\)
Now,
\(q\) is true, as \(3 \times 20=60\)
\(r\) is true, as \(5 \times 12=60\)
We know that for an ‘and’ statement to be true, both the component statements must be true.
Here, \(q\) and \(r\) are true.
Therefore, \(p: q \wedge r\) is also true.

Q.2. Prove that \(3+\sqrt{7}\) is irrational.
Sol:
Let’s prove by contradiction.
Step 1: Assume \(3+\sqrt{7}\) is rational.
Step 2: Prove the assumption.
Write \(3+\sqrt{7}\) as \(\frac{a}{b}\), where \(a\) and \(b\) are co-primes, and \(b \neq 0\)
\(\Rightarrow 3+\sqrt{7}=\frac{a}{b}\)
\(\sqrt{7}=\frac{a}{b}-3\)
\(\sqrt{7}=\frac{a-3 b}{b}\)
We know that \(\sqrt{7}\) is irrational, but \(\frac{a-3 b}{b}\) is rational.
Hence, rational \(\neq\) irrational
This is a contradiction.
Therefore, \(3+\sqrt{7}\) is rational.
Hence, proved.

Q.3. Using the method of contrapositive, prove the statement: If \(x\) is an integer and \(x^{2}\) is even, then \(x\) is even.
Sol:
Given: If \(x\) is an integer, and \(x^{2}\) is even, then \(x\) is even.
Let,
\(p\) : If \(x\) is an integer, \(x^{2}\) is even
\(q: x\) is even
Step1: Assume \(q\) is false.
\(\therefore x\) is not even
\(\Rightarrow x\) is odd
\(\therefore x=2 n+1\)
Step 2: Prove \(p\) is false.
\(\Rightarrow x^{2}\) is not even
\(\Rightarrow(2 n+1)^{2}\) is not even
\(\Rightarrow(2 n)^{2}+2(2 n)+1\)
\(\Rightarrow 4 n^{2}+4 n+1\)
\(\Rightarrow 4 n(n+1)+1\)
Any multiple of \(4\) is even, and when incremented by \(1\) is odd.
Step 3: Since \(p\) and \(q\) are false, the given statement is true.
Therefore, if \(x\) is an integer, and \(x^{2}\) is even, then \(x\) is even.

Q.4. Show the statement is false.
\(p\) : If \(n\) is an odd positive integer, then \(n\) is prime.
Sol:
Given:
\(p\) : If \(n\) is an odd positive integer, then \(n\) is prime.
Let,
\(q: n\) is an odd positive integer
\(r: n\) is prime
Let \(n=9\)
Now,
\(q\) is true, as \(9\) is an odd positive integer.
\(r\) is false, as \(9=3 \times 3\) and is not a prime.
Hence, \(p \Rightarrow q\) is not true.
Therefore, \(p\) is not true.

Q.5. Using a counter-example, show that the following statement is not true. \(p\) : The equation \(x^{2}-1=0\) does not have a root lying between \(0\) and \(2\).
Sol:
Given:
\(p\) : The equation \(x^{2}-1=0\) does not have a root lying between \(0\) and \(2\).
Counter-example: \(x^{2}-1=0\)
\(\Rightarrow(x+1)(x-1)=0\)
\(\therefore x=\pm 1\)
Note that one of the roots, \(x=1\), lies between \(0\) and \(2\).
Hence, \(p\) is not true.

Summary

Statements are sentences that possess a specific fact, opinion or idea. Every statement is either true or false, and it cannot be both. Validating statements helps us get the truth value of the statements. The various techniques to validate are discussed based on the types of sentences. The rules to validate specific logical operators such as and, or, if-then and if and only if. There are four methods that are used in validating statements. They are the direct method, contrapositive method, contradictive method, and using counter-examples. Examples, where the statement is not valid are called a counter-example.

Frequently Asked Questions (FAQs)

Q.1. What do statements mean in math?
Ans: Mathematically, statements are sentences that are either true or false, but not both. They usually carry a fact, opinion, or idea.
Examples:
1. \(25\) is a prime number
2. The multiples of \(6\) are divisible by \(2\) and \(3\).

Q.2. What are validating statements in mathematical reasoning?
Ans: As we know, statements are either true or false. There are several methods and rules that can be adapted to find the truth value of a given statement. This process is called validating statements.

Q.3. How do you check the validity of a statement in math?
Ans: Follow the steps to check the validity of a given statement.
Step 1: Identify the logical connector.
Step 2: Choose a method to solve the statement.
Step 3: Make assumptions or find the truth value of the component statements.
Step 4: Deduce the truth value of the given statement.

Q.4. How do you solve logic statements?
Ans: There are four methods to solve logical statements. They are:
(i) Direct method
(ii) Contrapositive method
(iii) Method of contradiction
(iv) Using counter-examples

Q.5. What is the validity of a statement?
Ans: The validity of a statement is the interval where the truth value of the given statement is true or the interval where the truth value of the statement is false.

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