Prove or refute:
- If R is reflexive than all subsets of R are reflexive.
This my solution:
Let {1,2,3) in R --> We know that R is reflexive -> R = {(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),*(3,3)}. We also know that R subset means: R is a set which all the elements are contained in another set, which means there are also reflexive.
Not sure if correct or totally wrong, please need help. Thank you very much.
I am afraid you are wrong. The answer is no. Counterexample is:
set $X = \mathbb{N}$, and $R\subseteq P(X\times X)$
$R = \lbrace(1,1), (1,2), (2,1), (2,2)\rbrace$.
Of course $R$ is reflexive, because $xRx$ holds for every $x\in X$.
Now we can take a subset of $R$, for example $R_1 = \lbrace(1,2), (2,1)\rbrace$.
It is easy to see that $R_1$ is not reflexive.