Explanation:
Set (mathematics)
A set is a collection of distinct
objects, considered as an object in its own right. Sets are one of the most
fundamental concepts in mathematics.
Definition
A set
is a gathering together into a whole of definite, distinct objects of our
perception and of our thought - which are called elements of the set.
The elements or members of a set can be
anything: numbers, people, letters of the alphabet, other sets, and so on. Sets
are conventionally denoted with capital letters. Sets A and B are equal if and only if they have precisely the same
elements.
Basic
operations
There are several fundamental operations for constructing new sets
from given sets.
Unions
The union of A and B,
denoted A ∪ B
Two sets can be "added"
together. The union of A and B,
denoted by A ∪ B, is the
set of all things which are members of either A or B.
Examples:
1. {1,
2} ∪ {red, white} ={1, 2, red, white}.
2. {1,
2, green} ∪ {red, white, green} ={1, 2, red, white, green}.
3. {1,
2} ∪ {1, 2} = {1, 2}.
(Our solved example in
mathguru.com uses this concept)
Some basic properties of unions:
1. A ∪ B = B ∪ A.
2. A ∪ (B ∪ C) = (A ∪ B) ∪ C.
3. A ⊆ (A ∪ B).
4. A ∪ A = A.
5. A ∪ ∅ = A.
6.
A ⊆ B if
and only if A ∪ B = B.
Intersections
A new set can also be constructed by determining which members two
sets have "in common". The intersection of A and B,
denoted by A ∩ B, is the set of all things which are
members of both A and B.
If A ∩ B = ∅, then A and B are said to be disjoint.
The intersection of A and B,
denoted A ∩ B.
Examples:
1. {1,
2} ∩ {red, white} = ∅.
2. {1,
2, green} ∩ {red, white, green} = {green}.
3. {1,
2} ∩ {1, 2} = {1, 2}.
(Our solved example in
mathguru.com uses this concept)
Some basic properties of intersections:
- A ∩ B = B ∩ A.
- A ∩ (B ∩ C) = (A ∩ B) ∩ C.
- A ∩ B ⊆ A.
- A ∩ A = A.
- A ∩ ∅ = ∅.
•
A ⊆ B if and
only if A ∩ B = A.
Complements
The relative
complement
of B in A
The complement of A in U
The symmetric
difference of A and B
Two sets can also be
"subtracted". The relative complement of B in A (also
called the set-theoretic difference of A and B),
denoted by A \ B (orA − B),
is the set of all elements which are members of A but not
members of B. Note that it is valid to "subtract" members
of a set that are not in the set, such as removing the element green from
the set {1, 2, 3}; doing so has no effect.
In certain settings all sets under
discussion are considered to be subsets of a given universal
set U. In such cases, U \ A is
called the absolute complement or simply complement of A,
and is denoted by A′.
Examples:
1. {1,
2} \ {red, white} = {1, 2}.
2. {1,
2, green} \ {red, white, green} = {1, 2}.
3. {1,
2} \ {1, 2} = ∅.
4. {1,
2, 3, 4} \ {1, 3} = {2, 4}.
5. If U is
the set of integers, E is the set of even integers, and O is
the set of odd integers, then E′ = O.
(Our solved example in mathguru.com
uses this concept)
Some basic properties of
complements:
1. A \ B ≠ B \ A.
2. A ∪ A′
= U.
3. A ∩ A′
= ∅.
4. (A′)′
= A.
5. A \ A =
∅.
6. U′
= ∅ and ∅′ = U.
7. A \ B = A ∩ B′.
http://en.wikipedia.org/wiki/Set_(mathematics)
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