Answer:
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 <em>NOT </em>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.