Answered

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

Sagot :

GinnyAnswer
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]
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