Validating Mathematical Statements

Mathematical statements of any form and nature can be validated as true or false based on logical operators, other statements, or conditions. Several aspects help validate whether a given Mathematical statement is true or false such as phrases like “and,” “our,” ” if-then,” “if only,” “for every – there exists, etc. Mathematical statement validations have several rules that help deduce whether a given statement is true or false. 

For example: If a statement says that a holds true if and only if b holds true, then the statement can be validated by knowing if b holds true as that will also prove that a is true.

Rules for validation of Mathematical statements:

There are several rules for Validating Mathematical Statements, such as:

The “and” statement:

In an “and” statement, two statements are in conjunction with the word “and.” For the whole statement to be true, both components of the statement individually need to be true. If both or any one of the components is false, then the whole statement is concluded to be false.

For example:

Statement: 2 is an even and prime number

Component 1 of the statement: 2 is an even number

Component 2 of the statement: 2 is a prime number

To validate that the statement is true, it is to be validated that 2 is even as well as prime.

2 is divisible by 2, and that makes 2 an even number

2 is divisible by itself and 1 and has no other factors; thus, it is a prime number as well.

Thus we can say that the statement is true as both the components of the statement are true.

The “or” statement:

In an “or” statement, any one of the components of the statement needs to be true for the complete statement to be true. Thus if a statement has two components connected with an “or,” then any one of the components being true will hold the entire statement true. Only if both components are false then the statement will hold false; for any other scenario, the statement will hold true.

For example:

Statement: 3 is an even number or an odd number

Component 1: 3 is an even number

Component 2: 3 is an odd number

Thus, if 3 is an even number, an odd number, or both, then the statement will hold true. Here 3 is an odd number, and thus the statement holds true.

The “If-Then” statements or Implication:

When a statement has components that are dependent on one another, i.e. if one component holds true only, then the other component will also hold true. The statement is framed with if and then phrases. For example, if 19 is an odd number then it is not divisible by 2.

There are two ways of validating these statements:

Direct Method:

When the first component is assumed to be true and then the second component is validated, it is known as the Direct Method of validation

Contrapositive Method:

When the second component of the statement is assumed to be false, and then the first component is proven to be false, it is known as the Contrapositive Method of validation.

For example:

Statement: If a is negative, then a2 is positive

Component 1: a is negative

Component 2: a2 is positive

Direct Method:

Considering a is positive, i.e. component 1 is true

a = -5

therefore, a2 = (-5)2 = 25 (hence component 2 is true)

The “if and only if” statement or bi-implication:

The statement “if and only if” is used as a bi-implication statement when one component of the statement will hold true if and only if the other component is true. Thus the validation of each component is dependent on the validation of the other component.

There are two validating Mathematical statement points for such statements:

Validating that if component 1 is true, then component 2 will also be true

Validating that if component 2 is true, then component 1 will also be true

For example:

Statement: If and only if 18 is divisible by 2, it will be an even number.

Component 1: 18 is divisible by 2

Component 2: 18 is an even number

If 18 is validated to be divisible by 2, which it is, then it is an even number.

Contradiction Method:

This is another validating Mathematical statement method. In this method, the given statement is assumed to be false and the contradicting statement or negation of the statement is assumed to be true. 

Let us use a counter example to validate this. 

(A counterexample is a kind of example where an invalid situation is taken into consideration)

Equation:

9×2 – 64 ≠ (3x-8)(3x+8)

Let us take x = 0

9(0)2 – 64 ≠ {(3*0)-8}{(3*0)+8}

-64 ≠ 64, which is a contradiction.

Thus the negation of the statement is wrong, and the statement itself is true.

Conclusion:

Therefore, we now know that every Mathematical statement can be validated to be true or false and that can be done using several methods. Each statement is broken into dependent components, and then one of the components is validated to be true or false, which then determines the validity of the other component. Thus in the case of Mathematical statements, the validation depends on both components. Some methods use the validation of one component as a validation of the other. Some methods have both components of the statement validating each other as those are dependent in both ways.