Answer :
Let’s analyze the given logical implications step by step:
1. We have the statement \( x \Rightarrow y \). This means if \( x \) is true, then \( y \) must also be true.
2. We also have the statement \( y \Rightarrow z \). This means if \( y \) is true, then \( z \) must also be true.
To determine which statement must be true among the options, let’s utilize the concept of the transitive property of logical implication:
- The transitive property of implication states that if \( x \Rightarrow y \) and \( y \Rightarrow z \), then we can conclude \( x \Rightarrow z \).
Now, let's revisit the options:
A. \( x \Rightarrow z \): From the transitive property of the given implications, we know that if \( x \Rightarrow y \) and \( y \Rightarrow z \), then \( x \Rightarrow z \) must be true. Therefore, this is a correct statement.
B. \( \neg x \Rightarrow \neg z \): This statement does not generally follow from the given implications. It requires additional negation rules and is not supported by the transitive property of implications.
C. \( \neg x \Rightarrow z \): This statement also does not generally follow from the given implications. There is no direct logical connection provided by the given information that leads to this statement.
D. \( z \Rightarrow x \): This statement represents the converse of what we are given which is not implied by the given conditions.
Based on our logical analysis, the correct answer is:
A. [tex]\( x \Rightarrow z \)[/tex].
1. We have the statement \( x \Rightarrow y \). This means if \( x \) is true, then \( y \) must also be true.
2. We also have the statement \( y \Rightarrow z \). This means if \( y \) is true, then \( z \) must also be true.
To determine which statement must be true among the options, let’s utilize the concept of the transitive property of logical implication:
- The transitive property of implication states that if \( x \Rightarrow y \) and \( y \Rightarrow z \), then we can conclude \( x \Rightarrow z \).
Now, let's revisit the options:
A. \( x \Rightarrow z \): From the transitive property of the given implications, we know that if \( x \Rightarrow y \) and \( y \Rightarrow z \), then \( x \Rightarrow z \) must be true. Therefore, this is a correct statement.
B. \( \neg x \Rightarrow \neg z \): This statement does not generally follow from the given implications. It requires additional negation rules and is not supported by the transitive property of implications.
C. \( \neg x \Rightarrow z \): This statement also does not generally follow from the given implications. There is no direct logical connection provided by the given information that leads to this statement.
D. \( z \Rightarrow x \): This statement represents the converse of what we are given which is not implied by the given conditions.
Based on our logical analysis, the correct answer is:
A. [tex]\( x \Rightarrow z \)[/tex].
