Check Whether the Statements are True or False
Question
Let \(R\) be a relation from \(N\) to \(N\) defined by
\[ R=\{(a,b):a,b\in N \text{ and } a=b^2\} \]
Are the following statements true?
(i) \[ (a,a)\in R \text{ for all } a\in N \]
(ii) \[ (a,b)\in R \Rightarrow (b,a)\in R \]
(iii) \[ (a,b)\in R \text{ and } (b,c)\in R \Rightarrow (a,c)\in R \]
Solution
(i)
For \[ (a,a)\in R, \] we must have \[ a=a^2 \]
This is not true for all natural numbers.
Hence,
\[ \boxed{\text{False}} \]
(ii)
If \[ (a,b)\in R, \] then \[ a=b^2 \]
For \[ (b,a)\in R, \] we need \[ b=a^2 \] which is not always true.
Hence,
\[ \boxed{\text{False}} \]
(iii)
If \[ (a,b)\in R, \] then \[ a=b^2 \]
and if \[ (b,c)\in R, \] then \[ b=c^2 \]
Therefore,
\[ a=(c^2)^2=c^4 \]
But for \[ (a,c)\in R, \] we need \[ a=c^2 \] which is not always true.
Hence,
\[ \boxed{\text{False}} \]