Question

Which of the following statements is the inverse of “If you do not understand geometry, then you do not know how to reason deductively”?
A. If you reason deductively, then you understand geometry.
B. If you understand geometry, then you reason deductively.
C. If you do not reason deductively, then you understand geometry.
D. None of these

Answer 1

Hint: We need to break the given condition or statement into two parts, hypothesis, and conclusion. We are trying to find the inverse statement. So, we take the inverse of both hypotheses and conclusions. The main theme of the statement doesn’t change as we have taken inverse of both parts. At last, we put two parts together to form the inverse statement.

Complete step-by-step solution:
To form the inverse of the conditional statement, we need to take the negation of both the hypothesis and the conclusion.
So, the form of the statement should be in the form ‘If p, then q’.
Here we can easily understand the two parts of the statement.
The ‘p’ is the hypothesis part which starts with a ‘if’.
The ‘q’ is the conclusion part which starts with a ‘then’.
Now, to find the inverse of the given statement we take the inverse of both parts, hypothesis, and conclusion.
So, the inverse of ‘If p, then q’ will be ‘If not p, then not q’.
For our statement “If you do not understand geometry, then you do not know how to reason deductively”, the ‘If you do not understand geometry’ part is the hypothesis part which we define as ‘p’ and ‘then you do not know how to reason deductively’ part is the conclusion part which we define as ‘q’.
The inverse of ‘p’ becomes ‘If you understand geometry’.
The inverse of ‘q’ becomes ‘then you know how to reason deductively’.
So, we got the inverse of two parts of the statement. Now, we put them together to find the final statement.
The final statement is “If you understand geometry, then you know how to reason deductively”.
This is the inverse statement of “If you do not understand geometry, then you do not know how to reason deductively”.
So, (B) is the correct option.

Note: We need to remember that when we are solving the inverse form of a statement, we aren’t changing the meaning of the statement. We are just changing the form which bears a similar meaning to the main statement. The inverse formula is independent of whether the parts of the statement are in a negative or affirmative sense. The only thing we need to do is to negate the parts.
We need to find the negation form of both the hypothesis and conclusion part as they cancel out each other to maintain the same meaning of the original statement. We can’t find the negation form of only 1 part.