# 3.1.3: Operations on sets

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Now that we have a way to express a wide variety of sets, we turn to operations that can be performed on sets. The most basic operations on sets are union and intersection. If*A** *and *B *are sets, then we define the union of *A *and *B *to be the set that contains all the elements of *A *together with all the elements of *B*. The union of *A *and *B *is denoted by*A** *∪ *B*. The union can be defined formally as

\(A \cup B=\{x | x \in A \vee x \in B\}\)

The intersection of *A *and *B *is defined to be the set that contains every entity that is both a member of *A *and a member of *B*. The intersection of *A *and *B *is denoted by *A*∩*B*. Formally,

\(A \cap B=\{x | x \in A \wedge x \in B\}\)

An entity gets into *A *∪ *B *if it is in either *A *or *B*. It gets into *A *∩ *B *if it is in both *A *and *B*. Note that the symbol for the logical ‘or’ operator, ∨, is similar to the symbol for the union operator, ∪, while the logical ‘and’ operator, ∧, is similar to the intersection operator, ∩.

The set **difference** of two sets, *A *and *B*, is defined to be the set of all entities that are members of *A *but are not members of *B*. The set difference of *A *and *B *is denoted *A *∖ *B *or alternatively as *A *− *B*. The idea is that *A *∖ *B *is formed by starting with *A *and then removing any element that is also found in *B*. Formally,

\(A>B=\{x | x \in A \wedge x \notin B\}\)

Union and intersection are clearly commutative operations. That is, *A *∪ *B *= *B *∪ *A *and*A** *∩ *B *= *B *∩ *A *for any sets *A *and *B*. However, set difference is not commutative. In general, *A*∖*B \(\neq\)* *B*∖*A*.

Suppose that \(A=\{a, b, c\}\), that \(B=\{b, d\},\) and that \(C=\{d, e, f\} .\) Then we can apply the definitions of union, intersection, and set difference to compute, for example, that:

*A*∪*B *= {*a*,*b*,*c*,*d*} *A*∩*B *= {*b*} *A*∖*B *= {*a*,*c*}

*A*∪*C *= {*a*,*b*,*c*,*d*,*e*, *f*} *A*∩*C *= ∅ *A*∖*C *= {*a*,*b*,*c*}

In this example, the sets *A *and *C *have no elements in common, so that *A *∩ *C *= ∅. There is a term for this: Two sets are said to be disjoint if they have no elements in common. That is, for any sets *A *and *B*, *A *and *B *are said to be disjoint if and only if *A** *∩ *B *= ∅.

Of course, the set operations can also be applied to sets that are defined by predicates. The next example illustrates this.

let *L*(*x*) be the predicate ‘*x *is lucky’, and let *W*(*x*) be the predicate ‘*x *is wise’, where the domain of discourse for each predicate is the set of people. Let*X** *= {*x *| *L*(*x*)}, and let *Y *= {*x *| *W*(*x*)}. Then

*X *∪ *Y *= {*x *| *L*(*x*) ∨ *W*(*x*)} = {people who are lucky or wise}

*X *∩ *Y *= {*x *| *L*(*x*) ∧ *W*(*x*)} = {people who are lucky and wise}

*X *∖ *Y *= {*x *| *L*(*x*) ∧ ¬*W*(*x*)} = {people who are lucky but not wise}

*Y *∖ *X *= {*x *| *W*(*x*) ∧ ¬*L*(*x*)} = {people who are wise but not lucky}

Figure 4.2: Some of the notations that are defined in this section. *A *and *B *are sets, and*a** *is an entity.

You have to be a little careful with the English word ‘and’. We might say that the set *X *∪ *Y *contains people who are lucky and people who are wise. But what this means is that a person gets into the set *X *∪ *Y *either by being lucky or by being wise, so *X *∪ *Y *is defined using the logical ‘or’ operator, ∨.