In many problems concerning discrete objects, it is often the case that there is some kind of relationship among the objects. Among a set of computer programs, we might say that two of the programs are related if they share some common data and are not related otherwise. Among a group of students, we might say two students are related if the first letters of their last names are the same. On the other hand, in a different situation we might want to say that two students are related if the first letters of their last names are different. Also, consider the set of integers {1, 2, 3 … 15}. We might say that three integers in the set are related if their sum is divisible by 5. Thus, the integers 2, 3, 5 are related and integers 5, 10, 15 are related, but the integers 1, 2, 4 are not.

We introduced the notion of an ordered pair of objects. Let A and B be two sets. The Cartesian product of A and B, denoted A × B, is the set of all ordered pairs of the form (a, b) where a ϵ A and b ϵ B. for example,

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

A binary relation from A to B is a subset of A × B. A binary relation is indeed only a formalization of the intuitive notion that some of the elements in A are related to some of the elements in B. As a matter of fact, if R is a binary relation from A to B and if the ordered pair (a, b) is in R, we would say that the element a is related to the element b. For example, let A = {a, b, c, d} be a set of six students, let B = {CS121, CS221, CS257, CS264, CS273, CS281} be a set of six courses. The Cartesian product A × B gives all the possible pairings of students and courses. On the other hand, a relation R = {(a, CS121), (b, CS221), (b, CS264,), (c, CS221), (c, CS257), (d, CS257), (d, CS281)} might describe the courses the students are talking, and a relation T = {(a, CS121), (c, CS257), (c, CS273)} might describe the courses the students are having difficulty with.

Besides a list of the ordered pairs, a binary relation can also be represented in tabular form or graphical form. For example, let A = {a, b, c, d} and B = , and let R = {a, ), (b, ), (c, ), (d, )} be a binary relation from A to B. R can be represented in tabular form, where the rows of the table correspond to the elements  in B, and a check mark in a cell means the element in the row containing the cell is related to the element in the column containing the cell. R can also be represented in graphical form, where the points in the left-hand column are the elements in A, the points in the right-hand column to a point in the right-hand column indicates that the corresponding element in A is related to the corresponding element in B.

Since binary relations are sets of ordered pairs, the notions of the intersection of two relations, the union of two relations, the symmetric difference of two relations, and the difference of two relations follow directly from that of sets. To be specific, let R₁ and R₂ be two binary relations from A to b. then R₁ ∩ R₂, R₁ ∪ R₂, R₁ R₂, and R₁ – R₂ are also binary relations from A to B, which are known as the intersection, the union, the symmetric difference, and the difference of R₁ and R₂. For example, let A = {a, b, c, d} be a set of courses. We might have a binary relation R₁ from A to B describing the courses that the students are taking and are also interested in.

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