If you are given a conditional statement that you know is true, can you predict whether
i) the converse is true?
ii) the inverse is true?
iii) the contrapositive is true?
no,no,yes
Thanks Steve!
To predict whether the converse, inverse, or contrapositive of a given conditional statement is true, you need to understand the logical structure of these statements.
Let's start by defining each of them:
1. Conditional Statement: A statement in the form "If A, then B." For example, "If it is raining, then the ground is wet."
2. Converse: The converse switches the order of the conditional statement, resulting in "If B, then A." For example, "If the ground is wet, then it is raining."
3. Inverse: The inverse negates both parts of the conditional statement, resulting in "If not A, then not B." For example, "If it is not raining, then the ground is not wet."
4. Contrapositive: The contrapositive flips and negates both parts of the conditional statement, resulting in "If not B, then not A." For example, "If the ground is not wet, then it is not raining."
Now, let's address each part of your question:
i) The converse of a true conditional statement is not guaranteed to be true. In other words, if you know that "If A, then B" is true, you cannot automatically predict that "If B, then A" is also true. Whether the converse is true or not depends on the actual relationship between A and B.
ii) Similarly, the inverse of a true conditional statement is not guaranteed to be true. Knowing that "If A, then B" is true does not allow you to predict whether "If not A, then not B" is true. Again, the truth of the inverse depends on the specific relationship between A and B.
iii) However, the contrapositive of a true conditional statement is always true. If you know that "If A, then B" is true, you can predict with certainty that "If not B, then not A" is also true. The contrapositive preserves the truth value of the original conditional statement.
To determine the truth value of the converse or inverse, you would need additional information or evidence to establish whether the relationships described are true or false. Always remember that the contrapositive is the only statement that is guaranteed to be true if the original conditional statement is true.