Applicable Mathematics/Sets

A set is a collection of items which have something in common. They are usually designated by upper case letters. The members of the set are called elements.

The number of elements in set A = n(A) or |A|

$$2\in A$$; 2 is an element of set A.

$$2\not\in C$$; 2 is not an element of set C.

$$B\subset A$$; B is a subset of A which means every element of B is an element of A. $$B\not\subset A$$; B is not a subset of A.

U; universal set and varies for each collection under construction.

$$\emptyset$$ is the empty set and has no elements in it. $$n(\emptyset) = 0.$$

Operations on Sets:

$$A\cap B$$; Intersection between sets: elements common to both sets.

$$A\cap B = \emptyset$$; A and B are said to be disjoint if they have no common elements.

$$A\cup B$$; The union between the sets A and B is the set of all the elements found in either A or B.

The complement of set A is the set of all elements in the universal set that are not found in A. A’ or. *Note: the complementation must be done in the context of the universal set.

For sets A and B; $$n(A\cup B) = n(A) + n(B) - n(A\cap B)$$

Venn Diagrams:

The relationship between sets and the operations performed on them can be represented graphically by using Venn Diagrams.

In Venn Diagrams, rectangles are usually used to represent universal sets and sets are usually defined on them represented by shaded circles or ovals.

Complement of a set:

Complement of set A is the set of all elements in the universal set NOT set A; A’ or.

Subsets:

Every element of a subset is also in the set. The set itself is also considered a subset. The empty set is also considered a subset.

Rules:

n(AB) = n(A) + n(B) – n(A  B)

n(ABC) = n(A) + n(B) + n(C) – n(A  B) – n(A  C) – n(B  C) + n(A  B  C)