Verifying The Statements
Verification of a mathematical statement means checking whether the statement is true or false by using logical reasoning, calculations, or examples. This process helps confirm the correctness of the statement before proving it formally.
We can verify different types of mathematical statements:
Direct Verification for Specific Cases
Example: "3 is a prime number"
Verification: Check if 3 has exactly two factors (1 and itself)
Process: List potential factors:
Only 1 and 3 divide 3 evenly, therefore
Counter-Example Method
Example: "All even numbers are composite"
To disprove, we just need one counter-example
Counter-example:
Therefore statement is
Algebraic Proof
Example: "For any real number x, 4x + x = 5x"
Verification through algebra:
Left side: 4x + x
Combine like terms: 4x + x =
Right side matches, therefore
Logical Deduction
Example: "4 and 5 are relative primes"
Verification: Find all factors of both numbers
Factors of 4:
Common factor is only
Therefore, they are relative primes.
Definition-Based Verification
Example: "A rhombus is a square"
Verify by comparing definitions:
Rhombus: Four equal
Since a rhombus doesn't necessarily have right angles, statement is
Historical/Factual Verification
Example: "Bhaskara has written a book 'Leelavathi'"
Requires historical research and documentation
Can be verified through historical records
Variable-Dependent Statements
Example: "x > 7"
Cannot be verified without specific value of x
Becomes verifiable only when x is defined
Always start with understanding the precise meaning of terms. Upon doing so, start by breaking complex statements into simpler parts. Make use of appropriate mathematical tools (algebra, geometry, etc.). Or look for counter-examples when trying to disprove. Always ensure all conditions in the statement are considered