Mathematical Relations
To understand the true meaning of the term relation, we have to review some concepts
from mathematics. Suppose that we have two sets, D1 and D2, where D1 = {2, 4} and D2 =
{1, 3, 5}. The Cartesian product of these two sets, written D1 × D2, is the set of all ordered
pairs such that the first element is a member of D1 and the second element is a member of
D2. An alternative way of expressing this is to find all combinations of elements with the
first from D1 and the second from D2. In our case, we have:
D1 × D2 = {(2, 1), (2, 3), (2, 5), (4, 1), (4, 3), (4, 5)}
Any subset of this Cartesian product is a relation. For example, we could produce a relation
R such that:
R = {(2, 1), (4, 1)}
We may specify which ordered pairs will be in the relation by giving some condition for
their selection. For example, if we observe that R includes all those ordered pairs in which
the second element is 1, then we could write R as:
R = {(x, y) | x ∈D1, y ∈D2, and y = 1}
Using these same sets, we could form another relation S in which the first element is
always twice the second. Thus, we could write S as:
S = {(x, y) | x ∈D1, y ∈D2, and x = 2y}
or, in this instance,
S = {(2, 1)}
since there is only one ordered pair in the Cartesian product that satisfies this condition.
We can easily extend the notion of a relation to three sets. Let D1, D2, and D3 be three sets.
The Cartesian product D1 × D2 × D3 of these three sets is the set of all ordered triples such
that the first element is from D1, the second element is from D2, and the third element is from
D3. Any subset of this Cartesian product is a relation. For example, suppose we have:
D1 = {1, 3} D2 = {2, 4} D3 = {5, 6}
D1 × D2 × D3 = {(1, 2, 5), (1, 2, 6), (1, 4, 5), (1, 4, 6), (3, 2, 5), (3, 2, 6), (3, 4, 5), (3, 4, 6)}
Any subset of these ordered triples is a relation. We can extend the three sets and define a
general relation on n domains. Let D1, D2, . . . , Dn be n sets. Their Cartesian product is
defined as:
D1 × D2 × . . . × Dn = {(d1, d2, . . . , dn) | d1 ∈D1, d2 ∈D2, . . . , dn ∈Dn}
and is usually written as:
Any set of n-tuples from this Cartesian product is a relation on the n sets. Note that in defining
these relations we have to specify the sets, or domains, from which we choose values.
To understand the true meaning of the term relation, we have to review some concepts
from mathematics. Suppose that we have two sets, D1 and D2, where D1 = {2, 4} and D2 =
{1, 3, 5}. The Cartesian product of these two sets, written D1 × D2, is the set of all ordered
pairs such that the first element is a member of D1 and the second element is a member of
D2. An alternative way of expressing this is to find all combinations of elements with the
first from D1 and the second from D2. In our case, we have:
D1 × D2 = {(2, 1), (2, 3), (2, 5), (4, 1), (4, 3), (4, 5)}
Any subset of this Cartesian product is a relation. For example, we could produce a relation
R such that:
R = {(2, 1), (4, 1)}
We may specify which ordered pairs will be in the relation by giving some condition for
their selection. For example, if we observe that R includes all those ordered pairs in which
the second element is 1, then we could write R as:
R = {(x, y) | x ∈D1, y ∈D2, and y = 1}
Using these same sets, we could form another relation S in which the first element is
always twice the second. Thus, we could write S as:
S = {(x, y) | x ∈D1, y ∈D2, and x = 2y}
or, in this instance,
S = {(2, 1)}
since there is only one ordered pair in the Cartesian product that satisfies this condition.
We can easily extend the notion of a relation to three sets. Let D1, D2, and D3 be three sets.
The Cartesian product D1 × D2 × D3 of these three sets is the set of all ordered triples such
that the first element is from D1, the second element is from D2, and the third element is from
D3. Any subset of this Cartesian product is a relation. For example, suppose we have:
D1 = {1, 3} D2 = {2, 4} D3 = {5, 6}
D1 × D2 × D3 = {(1, 2, 5), (1, 2, 6), (1, 4, 5), (1, 4, 6), (3, 2, 5), (3, 2, 6), (3, 4, 5), (3, 4, 6)}
Any subset of these ordered triples is a relation. We can extend the three sets and define a
general relation on n domains. Let D1, D2, . . . , Dn be n sets. Their Cartesian product is
defined as:
D1 × D2 × . . . × Dn = {(d1, d2, . . . , dn) | d1 ∈D1, d2 ∈D2, . . . , dn ∈Dn}
and is usually written as:
Any set of n-tuples from this Cartesian product is a relation on the n sets. Note that in defining
these relations we have to specify the sets, or domains, from which we choose values.
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