To determine which statement must be true, let's analyze the logical implications given in the question.
We are given two implications:
1. [tex]\(a \Rightarrow b\)[/tex]
2. [tex]\(b \Rightarrow c\)[/tex]
We need to find out which of the provided statements must necessarily follow from these implications.
Step-by-Step Solution:
1. Implication Connection:
- If [tex]\(a \Rightarrow b\)[/tex], it means whenever [tex]\(a\)[/tex] is true, [tex]\(b\)[/tex] must also be true.
- If [tex]\(b \Rightarrow c\)[/tex], it means whenever [tex]\(b\)[/tex] is true, [tex]\(c\)[/tex] must also be true.
2. Transitivity of Implication:
- By the transitivity property of logical implications, if [tex]\(a \Rightarrow b\)[/tex] and [tex]\(b \Rightarrow c\)[/tex], then [tex]\(a \Rightarrow c\)[/tex] follows logically. This means whenever [tex]\(a\)[/tex] is true, [tex]\(c\)[/tex] must also be true.
3. Analyzing the choices:
- A. [tex]\( \neg a \Rightarrow \neg c \)[/tex]:
This statement does not necessarily follow from the given implications. The negation of an implication does not preserve the original logical structure in a straightforward way.
- B. [tex]\( a \Rightarrow c \)[/tex]:
This statement follows directly from the transitivity of the given implications. If [tex]\(a \Rightarrow b\)[/tex] and [tex]\(b \Rightarrow c\)[/tex], then [tex]\(a \Rightarrow c\)[/tex] must be true.
- C. [tex]\( c \Rightarrow a \)[/tex]:
This statement suggests a reverse implication, which is not guaranteed from the given information. The given implications do not provide a reason for [tex]\(c\)[/tex] to imply [tex]\(a\)[/tex].
- D. [tex]\( \neg a \Rightarrow c \)[/tex]:
This statement does not necessarily follow from the given implications. The relationship between the negation of [tex]\(a\)[/tex] and [tex]\(c\)[/tex] is not something we can deduce from [tex]\(a \Rightarrow b\)[/tex] and [tex]\(b \Rightarrow c\)[/tex].
Given the logical analysis, the statement that must be true is:
B. [tex]\( a \Rightarrow c \)[/tex].
