Question

precision if i roll two 6 - sided dice and the sum of the numbers is 11, then one die must be a 5. write the converse, inverse, and contrapositive of the conditional statement. determine whether each related conditional is true or false. if a statement is false, then find a counterexample.
converse: select choice
select choice
then select choice
counterexample: i roll two 6 - sided dice and one die is a
5, then the sum of the numbers is
select choice 11, then one die
select choice a 5. the converse is
select choice
select choice
inverse: if i roll two 6 - sided dice and the sum of the numbers
select choice
counterexample: i roll two 6 - sided dice and the sum of the numbers
select choice a 5. the inverse is
select choice
select choice
contrapositive: if i roll two 6 - sided dice and one of the die
select choice a 5, then the sum
select choice 11. the contrapositive is
select choice
select choice

Explanation:

Step1: Identify the original statement

The original conditional statement is: "If I roll two 6 - sided dice and the sum of the numbers is 11, then one die must be a 5." Let's denote \(p\): "I roll two 6 - sided dice and the sum of the numbers is 11", \(q\): "one die must be a 5".

Step2: Recall the definitions of converse, inverse, and contrapositive

• Converse: The converse of \(p

ightarrow q\) is \(q
ightarrow p\). So, the converse should be: "If one die is a 5, then I roll two 6 - sided dice and the sum of the numbers is 11." But we know that if one die is 5, the other die can be from 1 to 6. For example, if one die is 5 and the other is 1, the sum is 6, not 11. So the converse is false. A counterexample is: I roll two 6 - sided dice and one die is a 5, but the sum is 6 (e.g., (5,1)).

• Inverse: The inverse of \(p

ightarrow q\) is \(
eg p
ightarrow
eg q\). So, the inverse is: "If I roll two 6 - sided dice and the sum of the numbers is not 11, then one die is not a 5." This is false. A counterexample: I roll two 6 - sided dice, the sum is 10 (e.g., (5,5)), here one die is 5 but the sum is not 11.

• Contrapositive: The contrapositive of \(p

ightarrow q\) is \(
eg q
ightarrow
eg p\). So, the contrapositive is: "If one die is not a 5, then I roll two 6 - sided dice and the sum of the numbers is not 11." Let's check. If one die is not 5, the maximum sum we can get is when the non - 5 die is 6 and the other die (not 5) is 6, sum is 12, but to get 11, we need 5 and 6. If there is no 5, we can't get 11 (since the possible pairs to get 11 are (5,6) and (6,5)). So the contrapositive is true.

Answer:

• Converse: If one die is a 5, then I roll two 6 - sided dice and the sum of the numbers is 11 (False, counterexample: (5,1) sum = 6).
• Inverse: If I roll two 6 - sided dice and the sum of the numbers is not 11, then one die is not a 5 (False, counterexample: (5,5) sum = 10).
• Contrapositive: If one die is not a 5, then I roll two 6 - sided dice and the sum of the numbers is not 11 (True).