Question

Write two other forms of each statement. All students who fight in school will be suspended.

Short answer

Contrapositive: If not suspended, then did not fight. Converse: If suspended, then fought.

Step-by-step solution

Identify the Logical Structure

The original statement is a conditional statement: "If a student fights in school, then that student will be suspended." The structure is "If a, then b," where 'a' is 'fights in school' and 'b' is 'will be suspended.'

Write the Contrapositive

The contrapositive of a conditional statement switches the hypothesis and conclusion and negates both. So, the contrapositive is: "If a student is not suspended, then that student did not fight in school."

Write the Converse

The converse of a conditional statement simply switches the hypothesis and the conclusion. Therefore, the converse is: "If a student is suspended, then that student fought in school."

Key concepts

Logical Structure

In the world of conditional statements, understanding the logical structure is crucial. This structure often takes the form of "If a, then b." Here, 'a' represents the hypothesis, and 'b' is the conclusion. This logical structure helps us understand how different statements are connected.

For example, in our exercise, the original statement is "If a student fights in school, then that student will be suspended." The statement is clear and organized in a straightforward "cause and effect" style. The fighting is the cause (hypothesis), and suspension is the effect (conclusion). This structure is fundamental in logic because it helps us carefully examine the relationship between different ideas or actions.

Breaking down statements in this way not only aids in critical thinking but also paves the way for understanding more complex logical concepts.

Contrapositive

The contrapositive is one of the fascinating components of conditional statements. It involves both switching the hypothesis and conclusion of the statement and negating them. This may sound a bit complicated at first, but let's simplify it.

For the statement "If a student fights in school, then that student will be suspended," the contrapositive would be "If a student is not suspended, then that student did not fight in school."

Notice how both parts of the statement were reversed and negated. Despite these changes, the truth value of the contrapositive is the same as the original statement. This means if the original statement is true, so is the contrapositive. Understanding contrapositives can help greatly in proving arguments and ensuring a deep comprehension of logical concepts.

Converse

The converse of a statement simply involves swapping the hypothesis and the conclusion. Let's take a closer look at how this applies to our example.

We started with a statement: "If a student fights in school, then that student will be suspended." To find the converse, we switch the two parts to create the statement: "If a student is suspended, then that student fought in school."

It's important to note that unlike the contrapositive, the truth value of the converse might not be the same as the original statement. In our example, it's not necessarily true that all suspended students fought—they could be suspended for other reasons. Therefore, understanding the converse helps highlight the differences between direct implications and assumptions made by reversing them.

Recognizing these differences is essential for evaluating logical arguments accurately.