Problem 35 - Thinking Mathematically

Question

Express each statement in "if ... then" form. (More than one correct wording in "if... then" form may be possible.) Then write the statement's converse, inverse, and contrapositive. Passing the bar exam is a necessary condition for being an attorney.

Step-by-Step Solution

Answer

The 'if...then' form is: 'If someone is an attorney, then they must have passed the bar exam.' The converse is: 'If someone had passed the bar exam, then they are an attorney.' The inverse is: 'If someone is not an attorney, then they have not passed the bar exam.' The contrapositive is: 'If someone has not passed the bar exam, then they are not an attorney.'

Step 1: Identify the Necessary Condition and Write it in 'If....Then' Form

The necessary condition for being an attorney is passing the bar exam. So, the statement in 'If... then' form is: 'If someone is an attorney, then they must have passed the bar exam.'

Step 2: Write the Statement's Converse

The converse of the 'if...then' statement swaps the 'if' portion with the 'then' portion. So the converse of the statement is: 'If someone had passed the bar exam, then they are an attorney.'

Step 3: Write the Statement's Inverse

The inverse of the 'if...then' statement negates both the 'if' portion and the 'then' portion. So the inverse of the statement is: 'If someone is not an attorney, then they have not passed the bar exam.'

Step 4: Write the Statement's Contrapositive

The contrapositive of the 'if...then' statement swaps the 'if' portion with the 'then' portion and negates both. So the contrapositive of the statement is: 'If someone has not passed the bar exam, then they are not an attorney.'

Key Concepts

If-Then Statements

If-then statements are fundamental in logical reasoning. They connect a condition to a result and are the basis for forming logical conclusions. In these statements, "if" represents a hypothesis or condition, while "then" introduces the outcome or consequence. For example, consider the statement, "If someone is an attorney, then they must have passed the bar exam." This communicates that being an attorney inevitably involves passing a specific exam.

When crafting if-then statements, it is crucial to clearly define both parts to avoid ambiguity. Many real-world situations can be expressed in this form, making them a powerful tool for argumentation and analysis. This structure helps in organizing thoughts, as well as in communicating complex relationships more effectively.

• "If" introduces a condition.
• "Then" shows the consequence of that condition.
• The statement connects two related ideas.

Converse Statements

A converse statement reverses the order of the hypothesis and the conclusion in an if-then statement. For instance, the converse of the statement "If someone is an attorney, then they must have passed the bar exam" is "If someone has passed the bar exam, then they are an attorney." Here, "passing the bar exam" becomes the condition and "being an attorney" becomes the result.

It's important to understand that the truth value of a converse statement may differ from the original statement. While the original may be true, its converse might not necessarily be true. In the previous example, just because someone has passed the bar exam doesn't automatically make them a practicing attorney.

• Converse statements flip the order of the original if-then components.
• The truth of the converse is not guaranteed by the truth of the original statement.

Inverse Statements

Inverse statements negate both the condition and the outcome of an original if-then statement. For example, the inverse of "If someone is an attorney, then they must have passed the bar exam" is "If someone is not an attorney, then they have not passed the bar exam." This structure tests the relationship by considering the absence or negation of the initial conditions.

Like converse statements, inverses can have different truth values than the original statements. The logical validity of the inverse isn't inherently carried over from the initial if-then format. Being conscious of this aspect is critical when employing inverse statements in logical arguments.

• Inverse statements negate both parts of the original.
• Their truth value is independent of the original statement.

Contrapositive Statements

Contrapositive statements are special because they maintain the truth value of the original if-then statement. They are formed by swapping and negating both the hypothesis and the conclusion of the original statement. For the statement "If someone is an attorney, then they must have passed the bar exam," the contrapositive is "If someone has not passed the bar exam, then they are not an attorney."

Unlike converses and inverses, contrapositive statements are logically equivalent to the original if-then statement. This means that if the original statement is true, the contrapositive is also true, and vice versa. This feature makes contrapositives uniquely useful in proofs and logical deductions.

• Contrapositives negate and flip both parts of the original statement.
• They are always logically equivalent to the original if-then statement.