Problem 28 - Thinking Mathematically

Question

Write the converse, inverse, and contrapositive of each statement. If all hard workers are successful, then some people are not hard workers.

Step-by-Step Solution

Answer

The converse of the statement is 'If some people are not hard workers, then all hard workers are successful'. The inverse is 'If some hard workers are not successful, then all people are hard workers'. The contrapositive is 'If all people are hard workers, then some hard workers are not successful'.

Step 1: Identify the Hypothesis and Conclusion

The statement 'If all hard workers are successful, then some people are not hard workers' could be written in the form 'If P, then Q', where 'P' is the hypothesis (all hard workers are successful) and 'Q' is the conclusion (some people are not hard workers).

Step 2: Form the Converse

The converse statement of 'If P, then Q' is 'If Q, then P'. Hence, the converse of the given statement is 'If some people are not hard workers, then all hard workers are successful'.

Step 3: Form the Inverse

The inverse statement of 'If P, then Q' is 'If not P, then not Q'. Therefore, the inverse of the statement is 'If some hard workers are not successful, then all people are hard workers'.

Step 3: Form the Contrapositive

The contrapositive of 'If P, then Q' is 'If not Q, then not P'. Thus, the contrapositive of the given statement is 'If all people are hard workers, then some hard workers are not successful'.

Key Concepts

Converse Statement

When we talk about converse statements, we're essentially flipping the hypothesis and conclusion of a given conditional statement. The form of a standard conditional statement can be described as: "If P, then Q." In this structure:

• P is the hypothesis or initial condition.
• Q is the conclusion drawn if P is true.

To create a **Converse Statement**, swap P and Q so that it reads: "If Q, then P." In the example, "If all hard workers are successful, then some people are not hard workers," the converse would be: "If some people are not hard workers, then all hard workers are successful." This new arrangement allows us to explore the logical relationships by testing if reversing the hypothesis and conclusion still holds true. Converse statements help us check for logical validity in different configurations and are useful in proofs and reasoning exercises.

Inverse Statement

An inverse statement takes a different approach by negating both the hypothesis and conclusion of the original conditional statement. Here's how it generally works:

• Begin with the same standard form: "If P, then Q."
• Negate both parts (P and Q) to get: "If not P, then not Q."

Applying this to our initial statement "If all hard workers are successful, then some people are not hard workers," the **Inverse Statement** becomes: "If some hard workers are not successful, then all people are hard workers." It's key to note that the truth of an inverse statement is independent from the original statement. Yet, constructing inverse statements allows for an exploration of what conditions might invalidate a hypothesis, boosting comprehension of logical structures and implications.

Contrapositive Statement

Contrapositive statements are incredibly important in logic because they always share the same truth value as the original conditional statement. This involves negating and swapping both the hypothesis and the conclusion:

• Starting with "If P, then Q,"
• Create a contrapositive by using "If not Q, then not P."

For the statement "If all hard workers are successful, then some people are not hard workers," the **Contrapositive Statement** becomes: "If all people are hard workers, then some hard workers are not successful." A contrapositive offers great insight because if the contrapositive is true, the original statement is also true, reinforcing the logical strength and validity of the argument. By understanding contrapositives, students develop a clearer sense of how negation and logical structuring overlap.