To solve this problem, we need to understand the implications and the logical structure involved.
We are given two implications:
1. [tex]\( x \Rightarrow y \)[/tex]
2. [tex]\( y \Rightarrow z \)[/tex]
We need to determine which statement logically follows from these two implications. Let’s analyze the statements one by one:
- A. [tex]\( x \Rightarrow z \)[/tex]
This statement suggests that if [tex]\( x \)[/tex] is true, then [tex]\( z \)[/tex] must be true. Given that [tex]\( x \Rightarrow y \)[/tex] and [tex]\( y \Rightarrow z \)[/tex], we can use the transitive property of implications. If [tex]\( x \)[/tex] leads to [tex]\( y \)[/tex], and [tex]\( y \)[/tex] leads to [tex]\( z \)[/tex], then logically [tex]\( x \)[/tex] should lead to [tex]\( z \)[/tex]. Therefore, [tex]\( x \Rightarrow z \)[/tex] is true.
- B. [tex]\( z \Rightarrow x \)[/tex]
This statement suggests that if [tex]\( z \)[/tex] is true, then [tex]\( x \)[/tex] must be true. There is no information given that supports a direct implication from [tex]\( z \)[/tex] back to [tex]\( x \)[/tex]. Hence, we cannot conclude [tex]\( z \Rightarrow x \)[/tex] based on the given information.
- C. [tex]\( \neg x \Rightarrow z \)[/tex]
This statement suggests that if [tex]\( x \)[/tex] is false, then [tex]\( z \)[/tex] must be true. There is no direct relationship or rule provided in the given information that supports this statement. Therefore, [tex]\( \neg x \Rightarrow z \)[/tex] cannot be concluded from the given premises.
- D. [tex]\( \neg x \Rightarrow \neg z \)[/tex]
This statement suggests that if [tex]\( x \)[/tex] is false, then [tex]\( z \)[/tex] must be false. This statement is related to the contrapositive, but the contrapositive of [tex]\( x \Rightarrow y \)[/tex] and [tex]\( y \Rightarrow z \)[/tex] would respectively be [tex]\( \neg y \Rightarrow \neg x \)[/tex] and [tex]\( \neg z \Rightarrow \neg y \)[/tex]. Hence, there is no direct relationship that confirms [tex]\( \neg x \Rightarrow \neg z \)[/tex] from the given premises.
Given the logical analysis of each statement, the only statement that must be true is:
A. [tex]\( x \Rightarrow z \)[/tex]
Thus, the correct answer is A.
