To solve this question, let's carefully analyze the given logical statements and their implications.
We are provided with two logical implications:
1. [tex]\( p \Rightarrow q \)[/tex]
2. [tex]\( q \Rightarrow r \)[/tex]
Our goal is to determine which statement must be true based on these implications.
### Step-by-Step Analysis:
Step 1: Understand the Meaning of Logical Implications
- [tex]\( p \Rightarrow q \)[/tex] means that if [tex]\( p \)[/tex] is true, then [tex]\( q \)[/tex] must also be true.
- [tex]\( q \Rightarrow r \)[/tex] means that if [tex]\( q \)[/tex] is true, then [tex]\( r \)[/tex] must also be true.
Step 2: Use Transitivity of Implications
We need to link these implications to find out a direct relationship between [tex]\( p \)[/tex] and [tex]\( r \)[/tex].
- Since [tex]\( p \Rightarrow q \)[/tex] and [tex]\( q \Rightarrow r \)[/tex], we can deduce that if [tex]\( p \)[/tex] is true, then [tex]\( q \)[/tex] must be true (by the first statement).
- Similarly, if [tex]\( q \)[/tex] is true, then [tex]\( r \)[/tex] must be true (by the second statement).
Combining these, we can conclude:
- If [tex]\( p \)[/tex] is true, then [tex]\( q \)[/tex] is true, and since [tex]\( q \)[/tex] is true, [tex]\( r \)[/tex] must also be true.
Thus, if [tex]\( p \)[/tex] is true, [tex]\( r \)[/tex] must be true. This gives us:
[tex]\[ p \Rightarrow r \][/tex]
### Step 3: Verify the Given Options
Now, let's check the given options to find the correct one:
- Option A: [tex]\( p \Rightarrow r \)[/tex]
- Option B: [tex]\( p \Rightarrow s \)[/tex]
- Option C: [tex]\( s \Rightarrow p \)[/tex]
- Option D: [tex]\( r \Rightarrow p \)[/tex]
From our deduction, we see that the true statement is:
[tex]\[ p \Rightarrow r \][/tex]
So, the correct answer is:
A. [tex]\( p \Rightarrow r \)[/tex]
