A conditional statement is always logically equivalent to its
a.) contrapositive
b.) converse
c.) conjunction
d.) inverse
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a. contrapositive
a.) contrapositive
To determine which option is correct, we need to understand the concepts of conditional statements and their logical equivalences.
A conditional statement is an "If-Then" statement where the "If" part is called the antecedent, and the "Then" part is called the consequent. It has the form "If A, then B," where A is the antecedent and B is the consequent.
The contrapositive of a conditional statement is formed by negating both the antecedent and the consequent and swapping their positions. For example, if we have the statement "If it is raining, then the ground is wet," the contrapositive would be "If the ground is not wet, then it is not raining."
The converse of a conditional statement is formed by swapping the positions of the antecedent and the consequent. Using the previous example, the converse would be "If the ground is wet, then it is raining."
The inverse of a conditional statement is formed by negating both the antecedent and the consequent without swapping their positions. With our example, the inverse would be "If it is not raining, then the ground is not wet."
Now, let's determine the correct answer:
a.) The contrapositive of a conditional statement is always logically equivalent. It has the same truth value as the original statement. Therefore, option a is a possible correct answer.
b.) The converse of a conditional statement is not always logically equivalent to the original statement. Swapping the positions of the antecedent and the consequent may result in a different truth value. Therefore, option b is not the correct answer.
c.) A conjunction is a different type of statement, where two or more statements are combined using the word "and." It is not directly related to the logical equivalence of a conditional statement. Therefore, option c is not the correct answer.
d.) The inverse of a conditional statement is also not always logically equivalent to the original statement. Negating both the antecedent and the consequent can change the truth value of the statement. Therefore, option d is not the correct answer.
Therefore, the correct answer is a.) the contrapositive.