Converse and Inverse

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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) $\neg p \to \neg q$
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?
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.
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
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?
6.
Original: If $x = 6$ , then $x^2 = 36$ . Find a counterexample to the converse, If $x^2 = 36$ , then $x = 6$ .
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
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?
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.
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
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?
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.
13.
Original: If $x > 5$ , then $x > 3$ . Check the converse, If $x > 3$ , then $x > 5$ , using whole numbers.
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
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.
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.
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.
18.
Is the biconditional $x^2 = 4x$ if and only if $x = 4$ true? Check both directions.