C, D are subsets of A.
How can i determine if these relations:
1)C X D union D X C
2)C X D intersection D X C
are reflexive symmetric or transitive relations?
Thanks for any kind of help
Hello rebbeca,
I will help you with the first one. Reflexivity in this case means $\displaystyle (\forall x\in A)[(x,x)\in (C\times D)\cup (D\times C)]$. Since $\displaystyle C$ and $\displaystyle D$ are subsets of $\displaystyle A$, they need not to be equal to $\displaystyle A$. This leads us to the idea that we may construct an counterexample to the statement of reflexivity.
Let $\displaystyle A=\{1,2,3\}$, $\displaystyle C=\{1\}$ and $\displaystyle D=\{2\}$. Then $\displaystyle C\times D=\{(1,2)\}$ and $\displaystyle D\times C=\{(2,1)\}$, thus $\displaystyle (C\times D)\cup (D\times C)=\{(1,2),(2,1)\}$. If we now take an arbitrary element of $\displaystyle A$, say $\displaystyle 1$, we see that $\displaystyle (1,1)$ is not an element of $\displaystyle (C\times D)\cup (D\times C)$. This is a rather extreme example, since also $\displaystyle (2,2)$ and $\displaystyle (3,3)$ are not elements of the relation $\displaystyle (C\times D)\cup (D\times C)$.
Symmetry in this case means $\displaystyle (\forall x,y\in A)[(x,y)\in (C\times D)\cup (D\times C) \rightarrow (y,x)\in (C\times D)\cup (D\times C)]$. Thus we start by letting $\displaystyle x,y$ be arbitrary elements of $\displaystyle A$ and assume $\displaystyle (x,y)\in (C\times D)\cup (D\times C)$. By the definition of union this means $\displaystyle (x,y)\in C\times D$ or $\displaystyle (x,y)\in D\times C$. Thus we have to consider two cases.
1. case: $\displaystyle (x,y)\in C\times D$
This means $\displaystyle x\in C$ and $\displaystyle y\in D$, thus $\displaystyle (y,x)\in D\times C$. We have therefore $\displaystyle (y,x)\in (C\times D)\cup (D\times C)$.
I hope my post helps you, so that you are able to do the second case on your own and to solve the problem.
Best wishes,
Seppel