Converse and Inverse

Show your work for each problem.

1.
What is the converse of the conditional $p \to q$ ?
1. a) $q \to p$
2. b) $\neg q \to \neg p$
3. c) $p \to \neg q$
4. d) $\neg p \to \neg q$
Answer: $q \to p$
The converse swaps the two parts: $q \to p$ . The statement $\neg p \to \neg q$ is the inverse, and $\neg q \to \neg p$ is the contrapositive, which you will meet in the next lesson.
2.
The statement If $x = 5$ , then $x^2 = 25$ is true. Its converse, If $x^2 = 25$ , then $x = 5$ , is false. Which negative number is a counterexample to the converse?
Answer: -5
$(-5)^2 = 25$ , so $x^2 = 25$ is true for $x = -5$ , but $x = 5$ is false. One counterexample shows the converse is false.
3.
Original: If $x = 2$ , then $3x = 6$ . Is the converse, If $3x = 6$ , then $x = 2$ , also true? Work it out step by step.
Answer: 1
1. Check the original: what is $3 \times 2$ ? 6
2. Solve $3x = 6$ . What is $x$ ? 2
3. How many values of $x$ solve $3x = 6$ ? 1
4.
What is the inverse of: If a shape is a square, then it is a rectangle?
1. a)If a shape is not a rectangle, then it is not a square
2. b)If a shape is a rectangle, then it is a square
3. c)If a shape is not a square, then it is not a rectangle
4. d)If a shape is a square, then it is not a rectangle
Answer: If a shape is not a square, then it is not a rectangle
The inverse negates both parts and keeps the order: If a shape is not a square, then it is not a rectangle. It is false: a 5 cm by 3 cm rectangle is not a square, but it is a rectangle.
5.
Converse: If a number is even, then it is divisible by 4. Which even number between 5 and 7 shows that this converse is false?
Answer: 6
6 is even, but $6 \div 4 = 1.5$ , so 6 is not divisible by 4. The hypothesis is true and the conclusion is false, so 6 is a counterexample.
6.
Original: If $x = 6$ , then $x^2 = 36$ . Find a counterexample to the converse, If $x^2 = 36$ , then $x = 6$ .
Answer: -6
1. What is $6^2$ ? 36
2. What is $(-6)^2$ ? 36
3. Which value of $x$ has $x^2 = 36$ but $x \neq 6$ ? -6
7.
Original: If a number is a multiple of 10, then it is a multiple of 5. Is the converse true or false?
1. a)False, because 20 is a multiple of 10
2. b)True, because the original statement is true
3. c)True, because every multiple of 5 is a multiple of 10
4. d)False, because 15 is a multiple of 5 but not of 10
Answer: False, because 15 is a multiple of 5 but not of 10
The converse is: If a number is a multiple of 5, then it is a multiple of 10. 15 is a multiple of 5, but $15 \div 10 = 1.5$ , so the converse is false. The truth of the original says nothing about the converse.
8.
Original: If a number is a multiple of 9, then it is a multiple of 3. Its converse is false. What is the smallest two-digit multiple of 3 that is not a multiple of 9?
Answer: 12
The two-digit multiples of 3 start with 12, 15, 18. Since $12 \div 9$ is not a whole number, 12 is a multiple of 3 but not of 9. It breaks the converse, If a number is a multiple of 3, then it is a multiple of 9.
9.
Original: If a number is divisible by 4, then it is divisible by 2. Test the inverse, If a number is not divisible by 4, then it is not divisible by 2, with the number 10.
Answer: 1
1. What is the remainder when 10 is divided by 4? 2
2. What is $10 \div 2$ ? 5
3. 10 is not divisible by 4 but is divisible by 2. How many counterexamples do you need to show the inverse is false? 1
10.
The converse of a conditional statement is false. What can you say about its inverse?
1. a)It is true
2. b)It can be true or false
3. c)It is false
4. d)It is the same as the original
Answer: It is false
The converse and the inverse always have the same truth value: their columns in the truth table are identical. So if the converse is false, the inverse is false too.
11.
In how many of the four rows of the truth table do $p \to q$ and its converse $q \to p$ have different truth values?
Answer: 2
$p \to q$ is false only in the row $p$ true, $q$ false. $q \to p$ is false only in the row $p$ false, $q$ true. In these 2 rows the values differ, in the other 2 rows both are true.
12.
Find the value of the converse $q \to p$ in the row where $p$ is false and $q$ is true. Write 1 for true and 0 for false.
Answer: 0
1. What is the value of $q$ in this row (1 or 0)? 1
2. What is the value of $p$ in this row (1 or 0)? 0
3. An if-then statement is false only when the hypothesis is true and the conclusion is false. What is $q \to p$ here (1 or 0)? 0
13.
Original: If $x > 5$ , then $x > 3$ . Check the converse, If $x > 3$ , then $x > 5$ , using whole numbers.
Answer: 2
1. What is the smallest whole number greater than 3? 4
2. Is $4 > 5$ ? Write 1 for yes and 0 for no. 0
3. How many whole numbers are greater than 3 but not greater than 5? 2
14.
Which statement can be written as a true biconditional (if and only if)?
1. a)If $x = 4$ , then $x^2 = 16$
2. b)If a shape is a square, then it has four sides
3. c)If a triangle has a $90^\circ$ angle, then it is a right triangle
4. d)If a number is a multiple of 4, then it is even
Answer: If a triangle has a $90^\circ$ angle, then it is a right triangle
Only the right triangle statement is true in both directions, because it is the definition of a right triangle. The others have false converses: $x = -4$ gives $x^2 = 16$ , 6 is even but not a multiple of 4, and a 5 cm by 3 cm rectangle has four sides but is not a square.
15.
Original: If $x > 4$ , then $x^2 > 16$ . Its converse is: If $x^2 > 16$ , then $x > 4$ . Find the integer counterexample to the converse that is closest to zero.
Answer: -5
A counterexample needs $x^2 > 16$ and $x \le 4$ . For $x = -4$ we get $(-4)^2 = 16$ , which is not greater than 16. For $x = -5$ we get $(-5)^2 = 25 > 16$ , but $-5$ is not greater than 4. So $-5$ is the counterexample closest to zero.
16.
Original: If a number is a multiple of 3, then it is a multiple of 6. Show that the original is false but the converse is true.
Answer: 2
1. What is the remainder when 9 is divided by 6? 3
2. 12 is a multiple of 6. What is $12 \div 3$ ? 4
3. Every multiple of 6 can be written as $6k = 3 \times 2k$ . What is $6 \div 3$ ? 2
17.
Compare the converse $q \to p$ and the inverse $\neg p \to \neg q$ in the truth table. Write 1 for true and 0 for false.
Answer: 4
1. Row $p = 1$ , $q = 0$ : what is $q \to p$ , that is $0 \to 1$ ? 1
2. Same row: $\neg p = 0$ and $\neg q = 1$ . What is $\neg p \to \neg q$ , that is $0 \to 1$ ? 1
3. Row $p = 0$ , $q = 1$ : what is $q \to p$ , that is $1 \to 0$ ? 0
4. Same row: $\neg p = 1$ and $\neg q = 0$ . What is $\neg p \to \neg q$ , that is $1 \to 0$ ? 0
5. In the rows $p = q = 1$ and $p = q = 0$ both are 1. In how many of the 4 rows do the converse and the inverse have the same value? 4
18.
Is the biconditional $x^2 = 4x$ if and only if $x = 4$ true? Check both directions.
Answer: 0
1. Direction If $x = 4$ , then $x^2 = 4x$ : what is $4^2$ ? 16
2. What is $4 \times 4$ , the value of $4x$ for $x = 4$ ? 16
3. Direction If $x^2 = 4x$ , then $x = 4$ : which other number solves $x^2 = 4x$ ? 0