Answer :
To address the question of which statement must be true given [tex]\(x \Rightarrow y\)[/tex] and [tex]\(y \Rightarrow z\)[/tex], we need to employ the transitive property of implications in logical reasoning. Let’s break it down step-by-step.
### Step-by-Step Solution:
1. Understand the Definitions and Premises:
- The implication [tex]\(x \Rightarrow y\)[/tex] means that if [tex]\(x\)[/tex] is true, then [tex]\(y\)[/tex] must also be true.
- Similarly, [tex]\(y \Rightarrow z\)[/tex] means that if [tex]\(y\)[/tex] is true, then [tex]\(z\)[/tex] must also be true.
2. Apply the Transitive Property:
- The transitive property in logic states that if [tex]\(A \Rightarrow B\)[/tex] and [tex]\(B \Rightarrow C\)[/tex], then [tex]\(A \Rightarrow C\)[/tex].
- Here, if we take [tex]\(x \Rightarrow y\)[/tex] as [tex]\(A \Rightarrow B\)[/tex] and [tex]\(y \Rightarrow z\)[/tex] as [tex]\(B \Rightarrow C\)[/tex], we can chain these implications together.
3. Chain the Implications:
- From [tex]\(x \Rightarrow y\)[/tex], we have that if [tex]\(x\)[/tex] is true, then [tex]\(y\)[/tex] is true.
- From [tex]\(y \Rightarrow z\)[/tex], we have that if [tex]\(y\)[/tex] is true, then [tex]\(z\)[/tex] is true.
- Combining these, if [tex]\(x\)[/tex] is true, then [tex]\(y\)[/tex] is true, and if [tex]\(y\)[/tex] is true, then [tex]\(z\)[/tex] is true. Therefore, if [tex]\(x\)[/tex] is true, [tex]\(z\)[/tex] must also be true.
4. Conclusion:
- Therefore, the combined implication gives us [tex]\(x \Rightarrow z\)[/tex].
Since [tex]\(x \Rightarrow z\)[/tex] must be true, the correct statement is:
[tex]\( \boxed{4} \)[/tex] or D. [tex]\( x \Rightarrow z \)[/tex]
### Step-by-Step Solution:
1. Understand the Definitions and Premises:
- The implication [tex]\(x \Rightarrow y\)[/tex] means that if [tex]\(x\)[/tex] is true, then [tex]\(y\)[/tex] must also be true.
- Similarly, [tex]\(y \Rightarrow z\)[/tex] means that if [tex]\(y\)[/tex] is true, then [tex]\(z\)[/tex] must also be true.
2. Apply the Transitive Property:
- The transitive property in logic states that if [tex]\(A \Rightarrow B\)[/tex] and [tex]\(B \Rightarrow C\)[/tex], then [tex]\(A \Rightarrow C\)[/tex].
- Here, if we take [tex]\(x \Rightarrow y\)[/tex] as [tex]\(A \Rightarrow B\)[/tex] and [tex]\(y \Rightarrow z\)[/tex] as [tex]\(B \Rightarrow C\)[/tex], we can chain these implications together.
3. Chain the Implications:
- From [tex]\(x \Rightarrow y\)[/tex], we have that if [tex]\(x\)[/tex] is true, then [tex]\(y\)[/tex] is true.
- From [tex]\(y \Rightarrow z\)[/tex], we have that if [tex]\(y\)[/tex] is true, then [tex]\(z\)[/tex] is true.
- Combining these, if [tex]\(x\)[/tex] is true, then [tex]\(y\)[/tex] is true, and if [tex]\(y\)[/tex] is true, then [tex]\(z\)[/tex] is true. Therefore, if [tex]\(x\)[/tex] is true, [tex]\(z\)[/tex] must also be true.
4. Conclusion:
- Therefore, the combined implication gives us [tex]\(x \Rightarrow z\)[/tex].
Since [tex]\(x \Rightarrow z\)[/tex] must be true, the correct statement is:
[tex]\( \boxed{4} \)[/tex] or D. [tex]\( x \Rightarrow z \)[/tex]
