Notions of Set

• Set: is a collection of elements
  • Capital letters (A, B, C . . .) are customarily used to symbolize sets.
• Elements: may be numbers, measurements, materials of any kind, plants, animals, men, or any other objects or entities.
  • Lowercase letters (a, b, c, . . .) are customarily used to symbolize elements.
• X = {a, b, c}: This is the way of denoting the set X.
  • Elements of a set are enclosed by braces, with com­mas in between.

Also

• X = {c, a, b}: The order of elements is not relevant.

Sets of Numbers:

Example 1:

S = {1, 3, 5, 7}: S is the set of positive, odd integers up to (and including) seven.

Example 2:

M = {1, 2, 3, 4, . . ., 50}: M is the set of all positive integers up to fifty.

Example 3:

N = {2, 4, 6, . . ., 24}:

N is the set of positive, even integers up to twenty- four.

Example 4:

Y = {5, 10, 15, 20, . . .}:

Y is the set of all positive integers, which are multiples of five.

• • When numbers are involved in a series, it is cus­tomary and desirable to list a sufficient number of elements, in a familiar order, so that the relation among numbers can be readily noticed.
• Finite set:

has limited numbers of elements, as X, S, M, and Nabove.

• Infinite set:

has an unlimited number of elements, as Y above.

• n(X):

is the way of denoting the number of elements in set X (which is 3, see above).

Example:

n(X = 3:

is the symbol for saying that set X has three elements. Also

n(S) = 4:(see above).

• • Null (or) empty set:

is a set with no elements, symbolized by 0.

Example:

H = Φ:

H is the set of horses that have horns.

Then n(H) = 0

Universal set:

is the largest set of a certain kind of elements, symbol­ized by U.

Example: All those who have visited Disneyland.

Subsets:  All other sets are subsets of U.

Examples:

All children who have visited Disneyland in 1999.

All newlyweds who have visited Disneyland so far.

• A = B

means that set A and set B contain exactly the same ele­ments.

Example:

A = {l, m, n, p}; B = {n, m, p, 1}

• A # B

means that sets of A and B do not contain exactly the same elements.

Example:

A = {s, r, i, r}; B = {s, l, i, n}

• Set A and set B are mutually exclusive (or disjoint) if they contain no common elements.

Example:

A = {v, 1, s, h, w}; B = {p, r, n, k}

• A C B

means that set A is a subset of set B.

Every element of set A is in set B.

Example:

A = {3, 4, 6, 7}; B = {3, 4, 6, 7, 12}

Every set is a subset of itself.

The null set is a subset of every set.

• A’ is the complement of set A.

A’ is the set of all elements in the universal set, except the elements in A.

Example: U = {1, 2, 3, 4, 6, 9}; A = {2, 4, 9}; A’ = {1, 3, 6}

Symbol for intersection of sets A and B

Set of all elements that belong to both A and B

Example: A El B = {a, f g, l}; A = {a, d, f g, l}; B = {a, c, f, g, l}

Symbol for the union of sets A and B

Set of all elements that belong to at least one of the two sets A and B

Example: A U B = {a, b, c, d, e, g, h}; A = {b, c, e, g}; B = {a, d, h}

Source: Srinagesh K (2005), The Principles of Experimental Research, Butterworth-Heinemann; 1st edition.