Certainly! Let's analyze the logical statements given:
1. [tex]\( a \Rightarrow b \)[/tex]
2. [tex]\( b \Rightarrow c \)[/tex]
We need to determine the logical conclusion that follows from these two statements.
### Step-by-Step Solution:
1. Understanding Implications:
- [tex]\( a \Rightarrow b \)[/tex]: If [tex]\( a \)[/tex] is true, then [tex]\( b \)[/tex] must also be true.
- [tex]\( b \Rightarrow c \)[/tex]: If [tex]\( b \)[/tex] is true, then [tex]\( c \)[/tex] must also be true.
2. Applying the Transitive Property of Implications:
- 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]. This means if the first condition implies the second, and the second condition implies the third, then the first condition must imply the third.
3. Deriving the Conclusion:
- From [tex]\( a \Rightarrow b \)[/tex] and [tex]\( b \Rightarrow c \)[/tex], we can deduce that [tex]\( a \Rightarrow c \)[/tex].
Therefore, based on the logical statements provided, the correct logical conclusion is:
[tex]\[ a \Rightarrow c \][/tex]
### Option Analysis:
- Option A: [tex]\( b \Rightarrow a \)[/tex] - This is incorrect because the original statements do not suggest that [tex]\( b \)[/tex] implies [tex]\( a \)[/tex].
- Option B: [tex]\( \neg b \Rightarrow a \)[/tex] - This is incorrect because none of the original statements or their contrapositive forms suggest that [tex]\( \neg b \)[/tex] implies [tex]\( a \)[/tex].
- Option C: [tex]\( \neg a \Rightarrow \neg c \)[/tex] - This is incorrect because the contrapositive of [tex]\( a \Rightarrow c \)[/tex] is actually [tex]\( \neg c \Rightarrow \neg a \)[/tex].
- Option D: [tex]\( a \Rightarrow c \)[/tex] - This is correct, as derived using the transitive property of logic.
Thus, the logical conclusion of the given conditional statements is:
[tex]\[ \boxed{\text{D. } a \Rightarrow c} \][/tex]
