Two subsets $A$ and $ B$ of the $(x,y)$ plane are said to be equivalent if there exists a function $f:A\to B$ which is both one-to-one and onto.
$(i)$ Show that two line segments in the plane are equivalent. $(ii)$ Show that any two circles in the plane are equivalent.
My solution goes like this:
$(i)$ Let $L_1$ and $L_2$ be two line segments in the plane. Without loss of generality, assume that $L_1$ and $L_2$ have the same length, and let $P_1Q_1$ and $P_2Q_2$ be the respective endpoints of $L_1$ and $L_2$. Let $f:L_1\to L_2$ be defined by $f(P_1)=P_2$ and $f(Q_1)=Q_2$. Then $f$ is clearly one-to-one and onto, and hence $L_1$ and $L_2$ are equivalent.
Is the solution for the first part correct? If not, why ? I dont have a clue about how to do the next part...
I struggled a lot with this question and only came to the conclusion that the equivalent property as described, is indeed an equivalent relation. But I dont have a clue about what to do next?
(i) Let $l$ and $m$ be two line segments in $xy$-plane. Let $C$ and $D$ be the set of points lying on line $l$ and $m$ respectively. Then clearly cardinality of both $C$ and $D$ is equal to the cardinality of the set of all real numbers. Hence $C$ and $D$ are equivalent sets.