Question

The inverse of the statement is “if you grew in Alaska, then you have seen snow.”
A. “If you did not grow up in Alaska, then you have not seen snow.”
B. “If you grow up in Alaska, then you have not seen snow.”
C. “If you did not grow up in Alaska, then you have seen snow.”
D. None of these.

Answer 1

Here, we will check which type of statement is given to us whose inverse has to find out. We will obtain that it is a conditional statement. Then we will express the given statement in the form of $p\to q$ where ‘p’ is the condition and ‘q’ is the result of that condition. Then, we will use the formula that inverse of a conditional statement represented by $p\to q$ is given by $\sim{p}\to \sim{q}$, and then we will write the negation of both ‘p’ and ‘q’ and write them together and hence we will get our required answer.

Complete step-by-step solution
Here, we need to find the inverse of the statement, “If you grew in Alaska, then you have seen snow.”
Now, we can see that this statement contains “if” and “then” and we know that if a statement contains “if” and “then”, then the statement is a conditional statement.
Hence, we need to find the inverse of the conditional statement.
Now, we know that a conditional statement comprises two parts, the condition (which starts from “if”) and the event as a result of that condition (which starts from “then”).
Now, let the condition be represented by ‘p’ and the event be ‘q’.
Thus, we can say that:
p=if you grew in Alaska
q=then you have seen snow
Now, we know that a conditional statement with the condition represented by ‘p’ and its result represented by ‘q’ is shown as:
$p\to q$
Now, the inverse of a conditional statement is also a conditional statement which comprises of the negation of both the parts of the statements written in the same way as the original statement, i.e. the inverse of the condition will still be the condition and the negation of the result will still be the result.
Symbolically, it can be shown as:
$\sim{p}\to \sim{q}$
Hence, we can say that:
$\text{Inverse of }p\to q=\sim{p}\to \sim{q}$
Here,
p=if you grew in Alaska
Thus, $\sim{p}=$ if you did not grow up in Alaska
q=then you have seen snow
Thus, $\sim{q}=$ then you have not seen snow
Hence, $\sim{p}\to \sim{q}$ can be written as:
“If you did not grow up in Alaska, then you have not seen snow.”
This is our required inverse of the given statement.
Hence, option (A) is the correct option.

Note: There are a few things that we should remember about a logical statement:
Firstly, we should know that the inverse of a logical statement is not equivalent to the original statement.
Secondly, we should know that there is a difference between the inverse of a conditional statement and the negation of a conditional statement. As mentioned above, the inverse of a conditional statement $p\to q$ is represented by: $\sim{p}\to \sim{q}$ and the negation of a conditional statement $p\to q$ is represented by $p\wedge \sim{q}$.