- 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.
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