To solve this problem, let's analyze the given logical implications and determine which statement must be true.
We are given two statements:
1. [tex]\( p \Rightarrow q \)[/tex] (If [tex]\( p \)[/tex] is true, then [tex]\( q \)[/tex] is true)
2. [tex]\( q \Rightarrow r \)[/tex] (If [tex]\( q \)[/tex] is true, then [tex]\( r \)[/tex] is true)
We need to evaluate the following options:
A. [tex]\( p \Rightarrow r \)[/tex]
B. [tex]\( s \Rightarrow p \)[/tex]
C. [tex]\( r \Rightarrow p \)[/tex]
D. [tex]\( p \Rightarrow s \)[/tex]
Let's consider each option in detail:
Option A: [tex]\( p \Rightarrow r \)[/tex]
1. Given [tex]\( p \Rightarrow q \)[/tex], if [tex]\( p \)[/tex] is true, then [tex]\( q \)[/tex] must be true.
2. Given [tex]\( q \Rightarrow r \)[/tex], if [tex]\( q \)[/tex] is true, then [tex]\( r \)[/tex] must be true.
From these two statements, we can link [tex]\( p \)[/tex] and [tex]\( r \)[/tex] using a logical transitivity property. If [tex]\( p \)[/tex] implies [tex]\( q \)[/tex] and [tex]\( q \)[/tex] implies [tex]\( r \)[/tex], then [tex]\( p \)[/tex] must imply [tex]\( r \)[/tex].
Thus, [tex]\( p \Rightarrow r \)[/tex] is a valid conclusion and must be true.
Option B: [tex]\( s \Rightarrow p \)[/tex]
There is no information given about [tex]\( s \)[/tex] and its relationship to [tex]\( p \)[/tex]. Therefore, we cannot conclude that [tex]\( s \Rightarrow p \)[/tex] must be true.
Option C: [tex]\( r \Rightarrow p \)[/tex]
This option suggests that if [tex]\( r \)[/tex] is true, then [tex]\( p \)[/tex] is true. However, there is no information given that directly relates [tex]\( r \)[/tex] back to [tex]\( p \)[/tex]. Hence, we cannot conclude that [tex]\( r \Rightarrow p \)[/tex] is true based on the given information.
Option D: [tex]\( p \Rightarrow s \)[/tex]
Similar to Option B, there is no information given about [tex]\( s \)[/tex] in relation to [tex]\( p \)[/tex]. Therefore, we cannot conclude that [tex]\( p \Rightarrow s \)[/tex] must be true.
After evaluating all the options, the statement that must be true given [tex]\( p \Rightarrow q \)[/tex] and [tex]\( q \Rightarrow r \)[/tex] is:
A. [tex]\( p \Rightarrow r \)[/tex]
