Validating Statements

In mathematical reasoning, we deal with different types of statements that may be true or false. We can say that the given statement is true based on the kinds of statements and the logical operator involved in it. That means, the given statement is true or not true is completely dependent upon which of the special words and phrases, such as “and”, “or”, and which of the implications “if and only”, “if-then”, and which of the quantifiers “for every”, “there exists”, appear in the given statement. Based on this, we can validate the statements given; also, we have different rules and techniques to validate the statements.

Validating Statements in Mathematical Reasoning

There are several ways to validate the given statements in mathematical reasoning. The important rules to check whether the given statement is true or false are given below.

Rule 1 Statements with “And”

If p and q are two mathematical statements, then to confirm that the statement “p and q” is true, the below steps must be followed.

Step 1: Prove that the statement p is true.

Step 2: Prove that the statement q is true.

Learn about compound statements here.

Rule 2 Statements with “Or”

If p and q are two mathematical statements, then to confirm that the statement “p or q” is true, one must examine the following two cases.

Case 1: By considering p is false, show that q must be true.

Case 2: By considering q is false, show that p must be true.

Rule 3 Statements with “If-then”

To verify the mathematical statement “if p then q,” we need to confirm that any one of the following cases is true.

Case 1: Taking p is true; show that q must be true. (Direct method)

Case 2: Taking q is false, show that p must be false. (Contrapositive method)

Click here to understand the use of if then statements in mathematical reasoning.

Rule 4 Statements with “if and only if ”

To verify the statement “p if and only if q”, we need to show the following:

(i) If p is true, then q is true

(ii) If q is true, then p is true

Thus, based on the given statement, we can apply the suitable rule from the above.

Go through the example given below to understand the application of the above defined rules to validate a given statement.

Solved Example

Question:

Check whether the following statement is true or not.

If x, y ∈ Z such that x and y are even numbers, then xy is even.

Solution:

Given statement is:

If x, y ∈ Z such that x and y are even numbers, then xy is even.

The components of this can be written:

p : x, y ∈ Z such that x and y are even

q : xy is even

To check the validity of the given statement, we apply Case 1 of Rule 3.

i.e. Taking p is true; show that q must be true.

So let us assume p is true that means x and y are even integers.

x = 2m for some integer m

y = 2n for some integer n

Now,

xy = (2m)(2n) = 2(2mn)

That means, xy is even.

Therefore, the given statement is true.

We can also use the contradiction method to validate whether the given statement is true or false. In the method of contradiction, to verify whether a given statement p is true, we assume that p is not true, i.e. ∼p (negation or inverse of p) is true. Later, we arrive at some result that contradicts our assumption. Therefore, we conclude that p is true. The method involves providing an example of a condition where the statement is not valid. Such an illustration is called a counter example. The name itself implies that this is an instance to counter the given statement. Also, counter examples are generally used to disprove the statement in mathematics. However, producing examples in support of a statement does not provide validity to the given statement.