let R be a relation on real numbers that is symmetric and transitive.
prove that if Dom(R) = real numbers then R is reflexive.
Can someone help me out with this please?
Hint: Can you make two steps and come to your initial position?
Of course, real numbers are irrelevant here. What matters is that for every $x$ there exists a $y$ such that $xRy$, i.e., $\operatorname{dom} R$ is the whole set on which $R$ is defined.
Proof:
Let R be a relation on R that is symmetric and transitive.
Suppose Dom(R) = R.
Consider real number x.
Since x is in Dom(R) = R, there exists y in R such that xRy.
Since R is symmetric, yRx.
Since R is transitive, xRx.
Thus, for all x in R, xRx.
Therefore, R is reflexive.