Let's analyze the given implications and apply the transitive property of logical implications to determine which statement must be true.
1. We are given two implications:
- [tex]\( a \Rightarrow b \)[/tex]
- [tex]\( b \Rightarrow c \)[/tex]
2. Using the transitive property of implication, if [tex]\( a \Rightarrow b \)[/tex] and [tex]\( b \Rightarrow c \)[/tex] hold true, then [tex]\( a \Rightarrow c \)[/tex] must logically follow. This is because:
- If [tex]\( a \)[/tex] is true, then [tex]\( b \)[/tex] must also be true (since [tex]\( a \Rightarrow b \)[/tex]).
- If [tex]\( b \)[/tex] is true, then [tex]\( c \)[/tex] must also be true (since [tex]\( b \Rightarrow c \)[/tex]).
- Therefore, if [tex]\( a \)[/tex] is true, then [tex]\( c \)[/tex] must also be true, leading to the conclusion [tex]\( a \Rightarrow c \)[/tex].
3. Let's examine the other options to ensure that none of them must be true based on the given information:
- B. [tex]\( c \Rightarrow a \)[/tex] is not necessarily true because we have no information that links [tex]\( c \)[/tex] directly back to [tex]\( a \)[/tex].
- C. [tex]\( \neg a \Rightarrow c \)[/tex] is not necessarily true based on the given implications. The negation of [tex]\( a \)[/tex] (i.e., [tex]\( \neg a \)[/tex]) does not provide information about [tex]\( c \)[/tex].
- D. [tex]\( \neg a \Rightarrow \neg c \)[/tex] is also not necessarily true. The negation of [tex]\( a \)[/tex] does not imply the negation of [tex]\( c \)[/tex]; there is no direct logical connection given between [tex]\( \neg a \)[/tex] and [tex]\( \neg c \)[/tex].
Hence, the statement that must be true is:
A. [tex]\( a \Rightarrow c \)[/tex]
Thus, the correct answer is:
A. [tex]\( a \Rightarrow c \)[/tex]
