1. Relations

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

2. Hello rebbeca,

I will help you with the first one. Reflexivity in this case means $(\forall x\in A)[(x,x)\in (C\times D)\cup (D\times C)]$. Since $C$ and $D$ are subsets of $A$, they need not to be equal to $A$. This leads us to the idea that we may construct an counterexample to the statement of reflexivity.

Let $A=\{1,2,3\}$, $C=\{1\}$ and $D=\{2\}$. Then $C\times D=\{(1,2)\}$ and $D\times C=\{(2,1)\}$, thus $(C\times D)\cup (D\times C)=\{(1,2),(2,1)\}$. If we now take an arbitrary element of $A$, say $1$, we see that $(1,1)$ is not an element of $(C\times D)\cup (D\times C)$. This is a rather extreme example, since also $(2,2)$ and $(3,3)$ are not elements of the relation $(C\times D)\cup (D\times C)$.

Symmetry in this case means $(\forall x,y\in A)[(x,y)\in (C\times D)\cup (D\times C) \rightarrow (y,x)\in (C\times D)\cup (D\times C)]$. Thus we start by letting $x,y$ be arbitrary elements of $A$ and assume $(x,y)\in (C\times D)\cup (D\times C)$. By the definition of union this means $(x,y)\in C\times D$ or $(x,y)\in D\times C$. Thus we have to consider two cases.

1. case: $(x,y)\in C\times D$

This means $x\in C$ and $y\in D$, thus $(y,x)\in D\times C$. We have therefore $(y,x)\in (C\times D)\cup (D\times C)$.

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

3. Thank you very much!