Establishing if a binary relation expressed in set-theoretical notation is antisymmetric

A relation R is said to be antisymmetric iff (i.e., if and only if), for any pair a; b for which a is in the relation R with b and b is in the relation R with a, then a and b are equal:

\[R\quad is\quad antisymmetric\quad
\Leftrightarrow\forall a,b \in A, (aRb\wedge bRa)\Rightarrow
(a=b)\]

Practical tip: You can spot an antisymmetric relation in its set-theoretical notation by checking that there are no repetitions of couples of elements (even if in a reversed order) and that, at the same time, there is at least one element in the relation with itself.

Consider the following relation:

cRb, aRa, dRb, dRa, bRb, dRc

On the set A = {a,b,c,d}

The relation can also be represented in a set-theorical notation:

R = {(c,b), (a,a), (d,b), (d,a), (b,b), (d,c)}

Exercise

Which of the following relations on the set A = {a, b, c, d} is actually an antisymmetric relation?

This is not the right answer. Remember: two elements are in a relation if their set-theoretical notation depicts them within round brackets. In an antisymmetric relation, if an element a is in a relation with b and, at the same time, b is in the same relation with a, then a=b. For example, an antisymmetric relation cannot have {(a,b) , (b,a)} in its set-theoretical notation.

This is not the right answer. Remember: two elements are in a relation if their set-theoretical notation depicts them within round brackets. In an antisymmetric relation, if an element a is in a relation with b and, at the same time, b is in the same relation with a, then a=b. For example, an antisymmetric relation cannot have {(a,b) , (b,a)} in its set-theoretical notation.

This is not the right answer. Remember: two elements are in a relation if their set-theoretical notation depicts them within round brackets. In an antisymmetric relation, if an element a is in a relation with b and, at the same time, b is in the same relation with a, then a=b. For example, an antisymmetric relation cannot have {(a,b) , (b,a)} in its set-theoretical notation.

Good job, you got it right! You're already on your way to becoming a mathematical psychologist!