Combination of Sets
We will know here that how sets can be combined in various ways to yield new sets. For example, let P be the set of students taking the course Theory of Computation and Q be the set of students taking the course Music Appreciation. If a certain announcement was made in both the theory of computation and the Music Appreciation classes, what is the set of students who know about the news announced? Clearly, it is the set of students who are taking either Theory of Computation or Music Appreciation, or both. If both these courses have their final examinations scheduled in the same hours, what is the set of students who will have conflicting final examinations? Clearly, it is the set of students taking both Theory of Computation and Music Appreciation. To finalize these notions, we define the union and the intersection of sets. The union of two sets P and Q, denoted P ∪ Q, is the set whose elements are exactly the elements in either P or Q (or both. For example,
{a, b} ∪ {c, d} = {a, b, c, d}
{a, b} ∪ {a, c} = {a, b, c}
{a, b} ∪ Ø = {a, b}
{a, b} ∪ {a, b} = {a, b, {a, b}}
The intersection of two points P and Q, denoted P ∩ Q, the set whose elements are exactly those elements that are in both P and Q. For example,
{a, b} ∩ {a, c} = {a}
{a, b} ∩ {c, d} = Ø
{a, b} ∩ Ø = Ø
If the elements in P are characterized by a common property and the elements in Q are characterized by another common property, then in the union of P and Q is the set of elements possessing at least one of these properties, and the intersection of P and Q is the set of elements possessing both of these properties. According to the definitions, P ∪ Q and Q ∪ P denote the same set, as do P ∩ Q and Q ∩ P.
The difference of two sets P and Q, denoted P – Q, is the set containing exactly those elements in P that are not in Q. For example,
{a, b, c} – {a} = {b, c}
{a, b, c} – {a, d} = {b, c}
{a, b, c} – {d, e} = {a, b, c}
The symmetric difference of two sets P and Q, denoted P Q, is the set containing exactly all the elements that are in P or Q but not in both. In other words, P Q is the set (P ∪ Q) – (P ∩ Q). for example,
{a, b} {a, c} = {b, c}
{a, b} Ø = {a, b}
{a, b} {a, b} = Ø
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