Discovering a Counterexample to Disprove a Conditional Statement in Mathematics

Counterexample of a conditional

A counterexample of a conditional statement is an example that proves the statement to be false

A counterexample of a conditional statement is an example that proves the statement to be false. It demonstrates that the given hypothesis does not necessarily imply the conclusion.

To understand this better, let’s define what a conditional statement is. A conditional statement is an “if-then” statement with two parts: the hypothesis (the “if” part) and the conclusion (the “then” part). The hypothesis is the initial condition or assumption, and the conclusion is the statement that follows from the assumption.

For example, consider the conditional statement: “If a number is divisible by 3, then it is an even number.”

To find a counterexample for this statement, we need to provide an example of a number that is divisible by 3 but is not an even number. Let’s take the number 9.

The hypothesis is that the number is divisible by 3, which is true for 9 as 9 ÷ 3 = 3. However, the conclusion states that the number is an even number, which is false for 9 since it is an odd number.

This counterexample disproves the conditional statement since we have found a number that satisfies the hypothesis but does not satisfy the conclusion. Hence, the statement “If a number is divisible by 3, then it is an even number” is false.

Finding counterexamples helps identify situations where a conditional statement does not hold true universally, making it a valuable tool in mathematical reasoning and proof-writing.

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