Question

“If Deb and Sam go to the mall then it is snowing” which statement below is logically equivalent?
A.If Deb and Sam do not go to the mall then it is not snowing
B.If Deb and Sam do not go to the mall then it is snowing
C.If it is snowing then Deb and Sam go to the mall
D.If it is not snowing Deb and Sam do not go to the mall

Answer 1

Hint: In this question we have given two propositions i.e. p and q, where
p: Dam and Sam go to the mall.
q: it is snowing.
Here both the statements are connected by “if-then” which is known as if-then implication and this implication is denoted by “$p \Rightarrow q$”. Apply the concept of if-then implication on the options and conclude the correct option.

Complete step-by-step answer:
When we declare something e.g. “Tom is a boy”. A sentence that makes a declaration is called a declarative sentence.
A declarative sentence which is either true or false can be called as a proposition. That means “Tom is a boy”, it can either be true or false but not both.
We have given two propositions and both are connected by if -then implication which is denoted by $p \Rightarrow q$.
In the given question, p: Dam and Sam go to the mall and q: it is snowing. We have the statement “If Deb and Sam go to the mall then it is snowing” it means p is true and q is true. Therefore $p \Rightarrow q$ is also true. It means that if one thing happens then other things will happen otherwise not.
Let us check each option one by one. In option A, the first statement is “If Deb and Sam do not go to the mall then it is not snowing”, it includes not p i.e. \[ \sim p\]and not q i.e. $ \sim q$. As, $ \sim p \Rightarrow \sim q$ is the inverse of $p \Rightarrow q$.
Let us check each option one by one. In option A, the statement is “If Deb and Sam do not go to the mall then it is snowing”, it includes not p i.e. \[ \sim p\] and not q i.e. $ \sim q$. As, $ \sim p \Rightarrow \sim q$ is the inverse of $p \Rightarrow q$.
Option B, the statement is “If Deb and Sam do not go to the mall then it is snowing”, it includes \[ \sim p\]and q. As, $ \sim p \Rightarrow q$is just converse of p.
Option C, the statement is “if it is snowing then Deb and Sam go to the mall”, it includes q and p. As, $q \Rightarrow p$ is converse of $p \Rightarrow q$.
Option D, the statement is “If it is not snowing Deb and Sam do not go to the mall”, it includes $ \sim q$ & $ \sim p$. As, $ \sim q \Rightarrow \sim p$ is contrapositive of $p \Rightarrow q$.
By seeing all the options, it is concluded that option D is correct as contrapositive statements are also equivalent statements.
So, option D is the correct answer.


Note: While solving this student confused in option C, they think option C has both positive statements while others consist of either one or both negative statements. This way is wrong, as q depends upon p but converse is not true.