The complement of a relation R, denoted as R̄ (or sometimes R'), is the set of all ordered pairs that are not in R, but are contained within the Cartesian product of the sets the relation is defined on.
Understanding the Complement of a Relation
To fully grasp the complement of a relation, consider a relation R defined from set A to set B. The Cartesian product, A × B, consists of all possible ordered pairs (a, b) where 'a' is an element of A and 'b' is an element of B.
The relation R is a subset of A × B. The complement of R, denoted as R̄, consists of all ordered pairs in A × B that are not in R. In other words:
R̄ = {(a, b) | (a, b) ∈ (A × B) and (a, b) ∉ R}
Example
Let's say we have two sets:
- A = {1, 2, 3}
- B = {x, y}
The Cartesian product A × B is:
A × B = {(1, x), (1, y), (2, x), (2, y), (3, x), (3, y)}
Now, let's define a relation R as:
R = {(1, x), (2, y), (3, x)}
The complement of R, R̄, would be:
R̄ = {(1, y), (2, x), (3, y)}
Notice that R̄ contains only the ordered pairs from A × B that are not present in R.
Key Takeaways
- The complement of a relation is defined relative to the Cartesian product of the sets involved.
- It includes all possible ordered pairs that could be in the relation (according to the Cartesian product) but are not.
- If an ordered pair is in the original relation, it is not in the complement, and vice versa.
In summary, the complement of a relation identifies all the ordered pairs that are possible but missing from the original relation within the context of the relevant sets.
