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If [tex]a \Rightarrow b[/tex] and [tex]b \Rightarrow c[/tex], which statement must be true?

A. [tex]\neg a \Rightarrow c[/tex]
B. [tex]c \Rightarrow a[/tex]
C. [tex]\neg a \Rightarrow \neg c[/tex]
D. [tex]a \Rightarrow c[/tex]


Sagot :

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]