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]$a \Rightarrow c$[/tex]
C. [tex]$c \Rightarrow a$[/tex]
D. [tex]$\neg a \Rightarrow \neg c$[/tex]
Answer (1)
The problem uses the transitive property of implication.
The transitive property states that if a A rr b and b A rrc , then a A rrc .
Applying this property to the given statements, we find that a A rrc must be true.
Therefore, the correct answer is a A rrc .
Explanation
Understanding the Problem We are given two conditional statements: a A rr b and b A rrc . We need to determine which of the given options must be true.
Applying Transitive Property The transitive property of implication states that if a A rr b and b A rrc , then a A rrc . This means that if a is true, then c must also be true.
Analyzing the Options Now, let's examine the given options:
A. ¬ a A rrc : This statement says that if a is not true, then c must be true. This is not necessarily true based on the given information. B. a A rrc : This statement says that if a is true, then c must be true. This is exactly what the transitive property tells us. C. c A rr a : This statement says that if c is true, then a must be true. This is not necessarily true. D. ¬ a A rr ¬ c : This statement says that if a is not true, then c is not true. This is not necessarily true.
Conclusion Based on the transitive property, the statement a A rrc must be true.
Examples
In computer science, if a program compiles ( a ) implies it has no syntax errors ( b ), and having no syntax errors ( b ) implies it can be executed ( c ), then compiling ( a ) implies it can be executed ( c ). This illustrates the transitive property in a practical context.
