To determine which statement must be true given that [tex]\( x \Rightarrow y \)[/tex] and [tex]\( y \Rightarrow z \)[/tex], let's analyze the implications step by step according to the properties of logical statements.
1. Given: [tex]\( x \Rightarrow y \)[/tex]
- This means if [tex]\( x \)[/tex] is true, then [tex]\( y \)[/tex] must also be true.
2. Given: [tex]\( y \Rightarrow z \)[/tex]
- This means if [tex]\( y \)[/tex] is true, then [tex]\( z \)[/tex] must also be true.
From the above two implications, we use the property of transitivity of implications:
- The transitivity of logical implications states that if [tex]\( x \Rightarrow y \)[/tex] and [tex]\( y \Rightarrow z \)[/tex], then [tex]\( x \Rightarrow z \)[/tex].
Let's analyze each of the given choices:
- A. [tex]\( \neg x \Rightarrow z \)[/tex]
- This statement is not necessarily true. From [tex]\( x \Rightarrow y \)[/tex] and [tex]\( y \Rightarrow z \)[/tex], we cannot deduce [tex]\( \neg x \Rightarrow z \)[/tex].
- B. [tex]\( x \Rightarrow z \)[/tex]
- This statement is true because, by transitivity, if [tex]\( x \Rightarrow y \)[/tex] and [tex]\( y \Rightarrow z \)[/tex], then [tex]\( x \Rightarrow z \)[/tex].
- C. [tex]\( \neg x \Rightarrow \neg z \)[/tex]
- This statement is not necessarily true either, as [tex]\( x \Rightarrow y \)[/tex] and [tex]\( y \Rightarrow z \)[/tex] do not imply [tex]\( \neg x \Rightarrow \neg z \)[/tex].
- D. [tex]\( z = x \)[/tex]
- This statement is not necessarily true. The implications [tex]\( x \Rightarrow y \)[/tex] and [tex]\( y \Rightarrow z \)[/tex] do not imply [tex]\( z \)[/tex] is equivalent to [tex]\( x \)[/tex].
Therefore, the correct statement that must be true given the initial conditions is:
- B. [tex]\( x \Rightarrow z \)[/tex].
