Find the best answers to your questions at Westonci.ca, where experts and enthusiasts provide accurate, reliable information. Explore thousands of questions and answers from knowledgeable experts in various fields on our Q&A platform. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.

If [tex]$x \Rightarrow y$[/tex] and [tex]$y \Rightarrow z$[/tex], which statement must be true?

A. [tex]\neg x \Rightarrow \neg z[/tex]
B. [tex]\neg x \Rightarrow z[/tex]
C. [tex]z \Rightarrow x[/tex]
D. [tex]x \Rightarrow z[/tex]


Sagot :

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]