What is the antisymmetric relations on the set {0,1}?
To find the antisymmetric relations on the set {0, 1}, we need to consider all possible pairs of elements from the set and determine if the relation satisfies the antisymmetric property.
An antisymmetric relation on a set states that if (a, b) is part of the relation, then (b, a) cannot be part of the relation, except for the case when a = b.
The set {0, 1} has four possible pairs: (0, 0), (0, 1), (1, 0), and (1, 1). Let's examine each pair:
1. (0, 0): Since a = b, this pair satisfies the condition for antisymmetry.
2. (0, 1): Here, (0, 1) is part of the relation. To check if (1, 0) is allowed, we look for the pair (1, 0) in the relation. If it is not present, then the relation is antisymmetric. In this case, (1, 0) is not part of the relation, so it satisfies the antisymmetric property.
3. (1, 0): Similarly, (1, 0) is part of the relation. Checking for (0, 1), if it is not present, then the relation is antisymmetric. Here, (0, 1) is not in the relation, so it is also antisymmetric.
4. (1, 1): Since a = b, this pair satisfies the condition for antisymmetry.
So, after examining all the pairs, we find that the antisymmetric relations on the set {0, 1} are:
- {(0, 0)}
- {(1, 1)}
- {(0, 0), (1, 1)}
Note that the empty set {} is also considered an antisymmetric relation since it vacuously satisfies the antisymmetry property.