let p be the relation defined on the set of all american citizens by xpy if and only if x and y are registered for the same political party. not being registered or being registered as an independent also counts as a registration. check all properties that p has.
anti-symmetric
reflexive
transitive
symmetric

This relation is symmetric, reflexive and transitive, but not anti-symmetric. Therefore it is an equivalence relation.

Step-by-step explanation:

Let's first prove that it is reflexive:

The explanation is as follows: let x be some american citizen, means that this person x is registered for the same political party as himself. This is obviously truth, because we are talking about the same person.

Next comes symmetry:

What does this statement mean? It means that if a is in the same party as b, then b is in the same party as a,  and viceversa. This must be true, for the statement tells us that x is in the same party as y, which can also be stated as "x and y are both in the same party".  This last statement also implies that y is in the same party as x, which is written as: . That proves that:

And the converse follows from the same reasoning.

Now for Transitivity:

What this statement means in this context is that if a,b and c are american citizens, and we have that it is simultaneously true that both a and b are in the same party, and that also b and c are in the same party, then a and c must be also in the same party. This is true because parties are exclusive organisations, you cannot be both a democrat and a republican at the same time, or an independent and  a republican. Therefore if a and b belong to the same party, and b and c also belong to the same party, it must be true that a belongs to the same party as b, and the same holds for c, therefore a and c belong to the same party (b's party). which we write as: . Thus it is true that R is a transitive relation.

Finally, Antisymmetry is NOT a property of this relation.

Let's see why, antisymmetry means:

That would mean that if x and y are two distinct american citizens , then if x is in the same party as y (), then it is not true that y is in the same party as x! ()

Clearly this isn't true, for example if x and y are two distinct democratic party members, we can say that that is, x and y are registered for the same party, and given that this relation is symmetric, as we have shown, we can also say , but this comes in conflict with the definition of antisymmetry. Thus we conclude that the relation R is not antisymmetric.

 On a final note, it's interesting to point out that reflexivity, symmetry and transitivity are the requirements for a relation to be an equivalence relation, which is a very useful concept in maths.