Answer :
To determine which statement must be true given the conditions [tex]\( a \Rightarrow b \)[/tex] and [tex]\( b \Rightarrow c \)[/tex], let's analyze each statement step-by-step.
1. Understand the implications:
- [tex]\( a \Rightarrow b \)[/tex] means that if [tex]\( a \)[/tex] is true, then [tex]\( b \)[/tex] must also be true.
- [tex]\( b \Rightarrow c \)[/tex] means that if [tex]\( b \)[/tex] is true, then [tex]\( c \)[/tex] must also be true.
2. Rewriting using logical equivalences:
- [tex]\( a \Rightarrow b \)[/tex] can be rewritten as [tex]\( \neg a \lor b \)[/tex].
- [tex]\( b \Rightarrow c \)[/tex] can be rewritten as [tex]\( \neg b \lor c \)[/tex].
3. Combining the statements:
- We have two statements: [tex]\( \neg a \lor b \)[/tex] and [tex]\( \neg b \lor c \)[/tex].
4. Analyze each option:
- Option A: [tex]\( \neg a \Rightarrow c \)[/tex]
- This can be rewritten as [tex]\( a \lor c \)[/tex].
- This statement is not necessarily true just because we have [tex]\( \neg a \lor b \)[/tex] and [tex]\( \neg b \lor c \)[/tex].
- Option B: [tex]\( c \Rightarrow a \)[/tex]
- This can be rewritten as [tex]\( \neg c \lor a \)[/tex].
- This statement is also not necessarily true given our initial implications.
- Option C: [tex]\( \neg a \Rightarrow \neg c \)[/tex]
- This can be rewritten as [tex]\( a \lor \neg c \)[/tex].
- This statement, like the previous ones, is not necessarily true given [tex]\( \neg a \lor b \)[/tex] and [tex]\( \neg b \lor c \)[/tex].
- Option D: [tex]\( a \Rightarrow c \)[/tex]
- This can be rewritten as [tex]\( \neg a \lor c \)[/tex].
- Now, let's check if this can be derived from the combined statements:
- We have [tex]\( \neg a \lor b \)[/tex].
- We also have [tex]\( \neg b \lor c \)[/tex].
- From [tex]\( \neg a \lor b \)[/tex], if [tex]\( a \)[/tex] is true, then [tex]\( b \)[/tex] must be true.
- If [tex]\( b \)[/tex] is true, from [tex]\( \neg b \lor c \)[/tex], [tex]\( c \)[/tex] must be true.
- Therefore, if [tex]\( a \)[/tex] is true, [tex]\( c \)[/tex] must be true, confirming that [tex]\( \neg a \lor c \)[/tex] is indeed true.
Thus, the statement [tex]\( a \Rightarrow c \)[/tex] (Option D) is logically consistent with the given premises [tex]\( a \Rightarrow b \)[/tex] and [tex]\( b \Rightarrow c \)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{a \Rightarrow c} \][/tex]
1. Understand the implications:
- [tex]\( a \Rightarrow b \)[/tex] means that if [tex]\( a \)[/tex] is true, then [tex]\( b \)[/tex] must also be true.
- [tex]\( b \Rightarrow c \)[/tex] means that if [tex]\( b \)[/tex] is true, then [tex]\( c \)[/tex] must also be true.
2. Rewriting using logical equivalences:
- [tex]\( a \Rightarrow b \)[/tex] can be rewritten as [tex]\( \neg a \lor b \)[/tex].
- [tex]\( b \Rightarrow c \)[/tex] can be rewritten as [tex]\( \neg b \lor c \)[/tex].
3. Combining the statements:
- We have two statements: [tex]\( \neg a \lor b \)[/tex] and [tex]\( \neg b \lor c \)[/tex].
4. Analyze each option:
- Option A: [tex]\( \neg a \Rightarrow c \)[/tex]
- This can be rewritten as [tex]\( a \lor c \)[/tex].
- This statement is not necessarily true just because we have [tex]\( \neg a \lor b \)[/tex] and [tex]\( \neg b \lor c \)[/tex].
- Option B: [tex]\( c \Rightarrow a \)[/tex]
- This can be rewritten as [tex]\( \neg c \lor a \)[/tex].
- This statement is also not necessarily true given our initial implications.
- Option C: [tex]\( \neg a \Rightarrow \neg c \)[/tex]
- This can be rewritten as [tex]\( a \lor \neg c \)[/tex].
- This statement, like the previous ones, is not necessarily true given [tex]\( \neg a \lor b \)[/tex] and [tex]\( \neg b \lor c \)[/tex].
- Option D: [tex]\( a \Rightarrow c \)[/tex]
- This can be rewritten as [tex]\( \neg a \lor c \)[/tex].
- Now, let's check if this can be derived from the combined statements:
- We have [tex]\( \neg a \lor b \)[/tex].
- We also have [tex]\( \neg b \lor c \)[/tex].
- From [tex]\( \neg a \lor b \)[/tex], if [tex]\( a \)[/tex] is true, then [tex]\( b \)[/tex] must be true.
- If [tex]\( b \)[/tex] is true, from [tex]\( \neg b \lor c \)[/tex], [tex]\( c \)[/tex] must be true.
- Therefore, if [tex]\( a \)[/tex] is true, [tex]\( c \)[/tex] must be true, confirming that [tex]\( \neg a \lor c \)[/tex] is indeed true.
Thus, the statement [tex]\( a \Rightarrow c \)[/tex] (Option D) is logically consistent with the given premises [tex]\( a \Rightarrow b \)[/tex] and [tex]\( b \Rightarrow c \)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{a \Rightarrow c} \][/tex]
