To determine which statement must be true given [tex]\( p \Rightarrow q \)[/tex] and [tex]\( q \Rightarrow r \)[/tex], we can use the transitive property of logical implications. Here are the steps to solve the problem:
1. Understanding the initial implications:
- The statement [tex]\( p \Rightarrow q \)[/tex] indicates that if [tex]\( p \)[/tex] is true, then [tex]\( q \)[/tex] must also be true.
- The statement [tex]\( q \Rightarrow r \)[/tex] indicates that if [tex]\( q \)[/tex] is true, then [tex]\( r \)[/tex] must also be true.
2. Applying the transitive property:
- The transitive property of implications tells us that if [tex]\( p \Rightarrow q \)[/tex] and [tex]\( q \Rightarrow r \)[/tex], then it must follow that [tex]\( p \Rightarrow r \)[/tex].
3. Checking each choice:
- Choice A: [tex]\( p = s \)[/tex]
- This statement is irrelevant to the given implications as it does not help us connect [tex]\( p \)[/tex] with [tex]\( r \)[/tex].
- Choice B: [tex]\( p = r \)[/tex]
- This statement set [tex]\( p \)[/tex] equal to [tex]\( r \)[/tex], but it is unnecessarily strong. We don't need equality; we just need an implication.
- Choice C: [tex]\( s = p \)[/tex]
- Again, this is irrelevant to the logic we are working with.
- Choice D: [tex]\( r = p \)[/tex]
- This implies [tex]\( p \)[/tex] and [tex]\( r \)[/tex] are the same, which was not established by the given implications.
4. Conclusion:
- The proper conclusion from the original statements [tex]\( p \Rightarrow q \)[/tex] and [tex]\( q \Rightarrow r \)[/tex] is [tex]\( p \Rightarrow r \)[/tex], which is most closely reflected in choice B when simplified conceptually.
Thus, the correct statement that must be true is:
B. [tex]\( p \Rightarrow r \)[/tex]
