Assume the following is a true statement.
If it is raining, I will carry an umbrella.
Which form of the original statement must also be true?
a- converse
b-- inverse
c- contrapositive
d- biconditional
My answer is c contrapositive
Right, if the statement is true, the contrapositive is true.
statement
If a cow, then a mammal
converse
If a mammal then a cow
who knows?
Inverse
If not a cow, then not a mammal
who knows?
Contrapositive
If not a mammal, then not a cow
TRUE
In other words if I am not carrying an umbrella, it is not raining.
The biconditional would be
I carry an umbrella if and only if it is raining.
However you might carry one even on a sunny day.
To determine which form of the original statement must also be true, let's review the definitions of each option:
a) Converse: Swapping the hypothesis and conclusion of the original statement. In this case, it would be "If I carry an umbrella, then it is raining."
b) Inverse: Negating both the hypothesis and conclusion of the original statement. This would lead to "If it is not raining, then I will not carry an umbrella."
c) Contrapositive: Negating and swapping the hypothesis and conclusion of the original statement. For our statement, the contrapositive would be "If I am not carrying an umbrella, then it is not raining."
d) Biconditional: It is a combination of the original statement and its converse. It is written as "If and only if" and indicates that both the original statement and its converse are true.
Now, the question asks which form must also be true. Since the original statement is assumed to be true, its contrapositive must also be true. Thus, your answer of c) contrapositive is correct.
To determine which form of the original statement must also be true, we need to understand the different forms of logical statements.
The original statement is an example of a conditional statement, which can be written in the form "If p, then q." In this case, p represents the condition "it is raining," and q represents the consequence "I will carry an umbrella."
Here are the different forms associated with a conditional statement:
a) Converse: This form switches the positions of p and q, resulting in "If q, then p." In this example, the converse would be "If I carry an umbrella, then it is raining."
b) Inverse: This form negates both p and q individually, resulting in "If not p, then not q." In this example, the inverse would be "If it is not raining, then I will not carry an umbrella."
c) Contrapositive: This form both switches the positions of p and q and negates them individually, resulting in "If not q, then not p." In this example, the contrapositive would be "If I am not carrying an umbrella, then it is not raining."
d) Biconditional: This form combines the original statement and its converse, stating that both "If p, then q" and "If q, then p" are true. It can be written as "p if and only if q." In this example, the biconditional would be "I will carry an umbrella if and only if it is raining."
Now, to identify which form must also be true, we need to consider the logical implications. In a conditional statement, the original statement and its contrapositive always have the same truth value. Therefore, if the original statement is true, the contrapositive must also be true.
Therefore, your answer is correct: the contrapositive is the form of the original statement that must also be true.