Operations on Sets

When we add, subtract, multiply, divide, or take the square root of some numbers, we are performing mathematical operations. Although sets are different from numbers, one can similarly perform certain mathematical operations on them. Three principal operations to be discussed here involve the union, intersection, and complement of sets.

To take the union of two sets A and B means to form a new set containing those elements (and only those elements) belonging to A, or to B, or to both A and B, The union set is symbolized by A U B (read: UA union B").

This example, incidentally, illustrates the ease in which two sets A and M are neither equal nor disjoint and in which neither is a subset oi the other.

Example 2

Again referring to Fig. 2.1, we see that the union of the set of all integers and the set of all fractions is the set of all rational numbers. Similarly, the union of the rational-number set and the irrational-number set yields the set of al! real numbers.

The intersection of two sets/! arid B, on the other hand, is a new set which contains those elements (and only those elements) belonging to both A and B. The intersection set is symbolized by /in/? (read: "A intersection 5").

Example 3

From the sets A and B in Example 1, we can write

Example 4 lf A = 6'1 and B = '9< 2> 7/ 4''then A n 8 = Set A and set B are disioint;there-- fore their intersection is the empty set—no element is common to A and B.

It is obvious that intersection is a more restrictive concept than union. In the former, only the elements common to A and B are acceptable, whereas in the latter, membership in either A or B is sufficient to establish membership in the union set. The operator symbols H and U—which, incidentally, have the same kind of general status as the symbols +. -=-, etc.—therefore have the connotations "and" and "or" respectively. This point can be better appreciated by comparing the following formal definitions of intersection and union:

FIGURE 2,2

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Union AUB

Intersection

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What about the complement of a set? To explain this, let us first introduce the concept of the universal set, In a particular contcxt of discussion, if the only numbers used are the set of the lirst seven positive integers, we may refer to it as the universal set U. Then, with a given set, say, A = {3, 6, 7}, wc can define another set A (read; "the complement of--4") as the set that contains all the numbers in the universal set U that arc not in the set A. That is,

Note that, whereas the symbol U has the connotation "or" and the symbol n means "and,*1 the complement symbol ^ carries the implication of "not11

Example 5 lf u = <5' 6'7' 8< 9>and A = t5< ^then * = i7' 8' 9l-

What is the complement of U7 Since every object (number) under consideration is included in the universal set, the complement of U must be empty. Thus U = 0.

The three types of set operation can be visualized in the three diagrams of Fig. 2.2, known as Venn diagrams. Tn diagram a, the points in the upper circle form a set A, and the points in the lower circle form a set B. The union of A and B then consists of the shaded area covering both circles, In diagram h are shown the same two sets (circles). Since their intersection should comprise only the points common to both sets, only the (shaded) overlapping portion of the two circlcs satisfies the definition. In diagram c, let the points in the rectangle be the universal set and la A be the set of points in the circle; then the complement set A will be the (shaded) area outside the circle.

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