Question

for 7 & 8, write the converse, inverse, and contrapositive of each true conditional statement. determine whether each statement is true or false. if a statement is false, find a counterexample

1. if you live in san diego, then you live in california.

converse:
inverse:
contrapositive:

1. if a polygon is a rectangle, then it is a square.

converse:
inverse:
contrapositive:

Explanation:

Step1: Analyze the original conditional statement

For problem 7: Original statement “If you live in San Diego, then you live in California.”
Let \(p\): “you live in San Diego” and \(q\): “you live in California”

Step2: Find the converse

The converse of \(p
ightarrow q\) is \(q
ightarrow p\). So the converse is “If you live in California, then you live in San Diego.” This is false. A counter - example: someone lives in Los Angeles (which is in California but not San Diego)

Step3: Find the inverse

The inverse of \(p
ightarrow q\) is \(
eg p
ightarrow
eg q\). The inverse is “If you do not live in San Diego, then you do not live in California.” This is false. Counter - example: lives in Los Angeles (not San Diego but in California)

Step4: Find the contrapositive

The contrapositive of \(p
ightarrow q\) is \(
eg q
ightarrow
eg p\). The contrapositive is “If you do not live in California, then you do not live in San Diego.” This is true. Because if you are not in California, you can't be in San Diego (which is a city in California)

For problem 8: Original statement “If a polygon is a rectangle, then it is a square.” Let \(p\): “a polygon is a rectangle” and \(q\): “it is a square”

Step5: Find the converse

The converse is \(q
ightarrow p\). “If a polygon is a square, then it is a rectangle.” This is true. By definition, a square is a special type of rectangle (a rectangle with all sides equal)

Step6: Find the inverse

The inverse is \(
eg p
ightarrow
eg q\). “If a polygon is not a rectangle, then it is not a square.” This is true. Because if it doesn't meet the criteria of a rectangle (four right angles), it can't be a square (which is a special rectangle)

Step7: Find the contrapositive

The contrapositive is \(
eg q
ightarrow
eg p\). “If a polygon is not a square, then it is not a rectangle.” This is false. Counter - example: a rectangle that is not a square (e.g., a rectangle with length \(l = 5\) and width \(w=3\))

Answer:

7.

• Converse: “If you live in California, then you live in San Diego.” False (Counter - example: Los Angeles).
• Inverse: “If you do not live in San Diego, then you do not live in California.” False (Counter - example: Los Angeles).
• Contrapositive: “If you do not live in California, then you do not live in San Diego.” True.

8.

• Converse: “If a polygon is a square, then it is a rectangle.” True.
• Inverse: “If a polygon is not a rectangle, then it is not a square.” True.
• Contrapositive: “If a polygon is not a square, then it is not a rectangle.” False (Counter - example: a non - square rectangle).