To solve the problem, we need to evaluate the given logical statements and determine which one must necessarily be true based on the conditions provided.
Given the conditions:
1. [tex]\( p \Rightarrow q \)[/tex]
2. [tex]\( q \Rightarrow r \)[/tex]
Let's analyze each of the possible statements:
Statement A: [tex]\( r \Rightarrow p \)[/tex]
This statement means that if [tex]\( r \)[/tex] is true, then [tex]\( p \)[/tex] must be true. However, this logical relationship does not follow from the given conditions. We cannot assert [tex]\( r \Rightarrow p \)[/tex] based on the information provided.
Statement B: [tex]\( p \Rightarrow r \)[/tex]
This statement means that if [tex]\( p \)[/tex] is true, then [tex]\( r \)[/tex] must be true. To confirm this, let's consider the given conditions. From [tex]\( p \Rightarrow q \)[/tex] and [tex]\( q \Rightarrow r \)[/tex], we see that if [tex]\( p \)[/tex] is true, then [tex]\( q \)[/tex] must be true (because [tex]\( p \Rightarrow q \)[/tex]). If [tex]\( q \)[/tex] is true, then [tex]\( r \)[/tex] must be true (because [tex]\( q \Rightarrow r \)[/tex]). By the transitive property of implication, this means [tex]\( p \Rightarrow r \)[/tex] is true. Therefore, statement B is correct.
Statement C: [tex]\( s \Rightarrow p \)[/tex]
This statement is about an unrelated variable [tex]\( s \)[/tex], which is not mentioned in the given conditions. We do not have any information about the relationship between [tex]\( s \)[/tex] and [tex]\( p \)[/tex]. Therefore, we cannot assert [tex]\( s \Rightarrow p \)[/tex] as necessarily true.
Statement D: [tex]\( p \Rightarrow s \)[/tex]
Similarly, this statement is about the relationship between [tex]\( p \)[/tex] and [tex]\( s \)[/tex]. Since the variable [tex]\( s \)[/tex] is not mentioned in the given conditions, we have no basis to conclude [tex]\( p \Rightarrow s \)[/tex]. Therefore, statement D is not necessarily true.
Based on our analysis, the only statement that must be true given the conditions is:
B. [tex]\( p \Rightarrow r \)[/tex]
