Prove That the Given Statement is Not Always True
Question
For the relation \(R_1\) defined on \(R\) by
\[ (a,b)\in R_1 \iff 1+ab>0 \]
Prove that
\[ (a,b)\in R_1 \text{ and } (b,c)\in R_1 \Rightarrow (a,c)\in R_1 \]
is not true for all \(a,b,c\in R\).
Solution
Take
\[ a=-2,\quad b=-\frac12,\quad c=1 \]
Now,
\[ 1+ab = 1+\left(-2\right)\left(-\frac12\right) \]
\[ =1+1=2>0 \]
Hence,
\[ (a,b)\in R_1 \]
Again,
\[ 1+bc = 1+\left(-\frac12\right)(1) \]
\[ =1-\frac12=\frac12>0 \]
Hence,
\[ (b,c)\in R_1 \]
Now,
\[ 1+ac = 1+(-2)(1) \]
\[ =1-2=-1<0 \]
Therefore,
\[ (a,c)\notin R_1 \]
Hence,
\[ \boxed{ (a,b)\in R_1 \text{ and } (b,c)\in R_1 \Rightarrow (a,c)\in R_1 \text{ is not always true.} } \]