Properties of the relation $R=A\times B \cup B\times A$

Question

A is a set. Let $B\subsetneq A$.

$R=A\times B \cup B\times A$

Determine if the relation is (a)reflexive, (b)symmetric, (c)transitive, (d)anti-reflexive, (e)anti-symmetric, (f)asymmetric, (g)equivalence relation.

This is what I did:

It isn't reflexive because there can't be a set that is a proper subset of itself. So it is anti-reflexive.

It is symmetric because a union is symmetric: $A\times B \cup B\times A=B\times A\cup A\times B$

It isn't transitive because not every ordered pair of (A and B) and (B and C) is in A and C, example:

$A=\{1\} \ B=\{1,2\} \ C=\{1,2,3\} \\ A\times B \cup B\times A = (1,1),(1,2),(2,1) \\ C\times B \cup B\times C = (1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2) \\ A\times C \cup C\times A = ( 1,1 ),( 1,2 ),(1 ,3 ),( 2,1 ),( 3,1 ) $

There is no equivalence relation.

Is it correct ? Thanks.

Answer 1

You need your counterexample to transitivity to be such that $B\subsetneq A$ and $C \subsetneq B$: $$C\subsetneq B \subsetneq A$$

So, for example, in your case, you'd need $$A = \{1, 2, 3\}, B = \{1, 2\}, C = \{1\}$$ Your approach, however, should still hold.

But you should note that because of the failure of reflexivity, the relation is not an equivalence relation, anyway.

Answer 2

You are asking about properties of $R=(A\times B)\cup(B\times A)$.

You know that $B\subsetneq A$, i.e., there is at least one element in $A\setminus B$. So let us choose one such element and denote it by $a_0\in A\setminus B$.

The relation $R$ is not reflexive, since $(a_0,a_0)\notin R$. (Can you explain why? Does the ordered pair $(a_0,a_0)$ belong to $A\times B$? Does it belong to $B\times A$?)

The relation $R$ is symmetric. (If $(x,y)\in R$ then the pair $(x,y)$ belongs to $A\times B$ or it belongs to $B\times A$. Suppose that $(x,y)\in A\times B$. What can you say about $(y,x)$?)

If the set $B$ has at least one element, then $R$ is not transitive. (Let $b_0\in B$. Then both $(a_0,b_0)$ and $(b_0,a_0)$ belong to $R$. Can you get from this a contradiction with transitivity?)

If the set $B$ has at least one element, then $R$ is not anti-reflexive. (Does $(b_0,b_0)$ belong to $R$?)

If the set $B$ has at least one element, then $R$ is not asymmetric. (Use that $(a_0,b_0)\in R$.)

If the set $B$ has at least one element, then $R$ is not antisymmetric. (Notice that $(a_0,b_0),(b_0,a_0)\in R$.)

Since $R$ is not reflexive, it is not an equivalence relation.