Question

If \(x=5\) and \(y=-2\) then \(x-2 y=9\). The contrapositive of this statement is (1) If \(x-2 y=9\) then \(x=5\) and \(y=-2\) (2) If \(x-2 y \neq 9\) then \(x \neq 5\) and \(y \neq-2\) (3) If \(x-2 y \neq 9\) then \(x \neq 5\) or \(y \neq-2\) (4) If \(x-2 y \neq 9\) then either \(x \neq 5\) or \(y=-2\)

Short answer

Option (3)

Step-by-step solution

Understand the Original Statement

The original statement says that if two conditions are true (x=5 and y=-2), then another condition (x−2y=9) is also true.

Identify the Contrapositive

The contrapositive of a statement of the form 'If P, then Q' is 'If not Q, then not P'. Recognize that 'not Q' means the negation of condition Q and 'not P' means the negation of conditions P.

Negate the Conclusion

Negate the conclusion x-2y=9 to get x-2y eq 9. This forms the 'not Q' part of the contrapositive.

Negate the Hypotheses

Each hypothesis also needs to be negated. Since the original hypotheses are x=5 and y=-2, their negations are x eq 5 and y eq -2.

Combine Negations in a Logical Statement

Combine the negations from steps 3 and 4, resulting in: 'If x-2y eq 9, then x eq 5 or y eq -2.' This follows logical principles that 'not A and not B' can be rewritten as 'not (A or B)'.

Match with Given Options

Review the given options and match the formed contrapositive: 'If x-2y eq 9 then x eq 5 or y eq -2' matches with option 3.

Key concepts

Logical Statements

Logical statements play a crucial role in mathematics and critical thinking. These statements are often expressed in the form 'If P, then Q,' where 'P' is a hypothesis and 'Q' is a conclusion. Understanding logical statements is essential for constructing valid arguments and performing mathematical proofs.

To illustrate, consider the original statement from the exercise: 'If x=5 and y=-2, then x-2y=9.' Here, the conditions x=5 and y=-2 form the hypothesis (P), and the equation x-2y=9 forms the conclusion (Q). Logical reasoning helps us determine whether the hypothesis leads to the conclusion.

Constructing the contrapositive involves reversing and negating these statements, ensuring the argument remains logically equivalent, which is a powerful tool in mathematical reasoning and proof.

Mathematical Reasoning

Mathematical reasoning involves using logical steps to solve problems and prove statements. This process often includes identifying patterns, making conjectures, and providing proofs. It helps establish the validity of mathematical statements through systematic analysis.

In our exercise, we use reasoning to convert the original statement to its contrapositive. Here’s how we systematically approach it:

• Identify the original statement: 'If x=5 and y=-2, then x-2y=9.'
• Understand logical equivalence: The contrapositive preserves the truth value of the original statement.
• Negate the components individually: The hypothesis 'x=5 and y=-2' becomes 'x≠5 or y≠-2.'
• Negate the conclusion: The conclusion 'x-2y=9' becomes 'x-2y≠9.'
• Combine the negated forms: This creates the contrapositive statement 'If x-2y≠9, then x≠5 or y≠-2.'

Through these steps, we see how mathematical reasoning enables us to rigorously convert and verify logical statements.

Negation

Negation is a fundamental concept in logic and mathematics. It involves creating the opposite of a given statement. When negating a statement, we deny its truth, which helps in forming contrapositives and tackling logical arguments.

In our exercise, negation plays a key role. The original statement is: 'If x=5 and y=-2, then x-2y=9.' To find the contrapositive, we first negate the conclusion and hypothesis:

• Negate the conclusion 'x-2y=9' to become 'x-2y≠9'
• Negate the hypotheses 'x=5 and y=-2' to 'x≠5 or y≠-2.' This uses the principle that 'not (A and B)' is 'not A or not B.'

Combining these negations forms the contrapositive: 'If x-2y≠9, then x≠5 or y≠-2.

Understanding how to negate statements correctly is essential as it allows us to reason about the truth and validity of logical statements systematically.