To solve the problem, let's carefully examine the given logical implications and deduce which statement must be true based on them:
1. We are given that [tex]\( a \Rightarrow b \)[/tex]. This means that if [tex]\( a \)[/tex] is true, then [tex]\( b \)[/tex] must also be true.
2. We are also given that [tex]\( b \Rightarrow c \)[/tex]. This means that if [tex]\( b \)[/tex] is true, then [tex]\( c \)[/tex] must also be true.
Now, let's use these two implications together:
- If [tex]\( a \)[/tex] is true (given by [tex]\( a \Rightarrow b \)[/tex]), [tex]\( b \)[/tex] must be true.
- Since [tex]\( b \)[/tex] is true (derived from the first implication), [tex]\( c \)[/tex] must also be true (from [tex]\( b \Rightarrow c \)[/tex]).
Thus, if [tex]\( a \)[/tex] is true, then [tex]\( b \)[/tex] is true, and if [tex]\( b \)[/tex] is true, then [tex]\( c \)[/tex] is true. By transitivity, we can infer that if [tex]\( a \)[/tex] is true, then [tex]\( c \)[/tex] must also be true. This combined implication is [tex]\( a \Rightarrow c \)[/tex].
Therefore, the statement [tex]\( a \Rightarrow c \)[/tex] must be true.
Correct answer:
D. [tex]\( a \Rightarrow c \)[/tex]
