SETS
SET
A Set is a well-defined collection of objects or elements or members.SOME SETS WHICH ARE USED PARTICULARLY IN MATHEMATICS
1. N: the set of all natural numbers.
2. Z: the set of all integers.
3. Q: the set of all rational numbers.
4. R: the set of all real numbers.
5. Z+: the set of positive integers.
6. Q+: the set of positive rational numbers
7. R+: the set of positive real numbers.
WAYS TO REPRESENT A SET
1. Roster or Tabular form: In roster or tabular form all the elements of the sets are written being separated by comma. For example, N = {1, 2, 3, 4, 5, 6, 7, 8, 9, …………}.
2. Set-Builder form: All the elements of a set possess a particular property(s). This property(s) is used to tell about a particular property. For example, N = {x: x is a natural number}. This is read as ‘the set of x such that x is a natural number’. Here colon is read as ‘such that and brasses as ‘the set of’.
TYPES OF SETS
1. Null/Empty/Void set: A set which does not contain any element is called the empty set or the null set or the void set. It is represented by Φ,
2. Singleton set: A set containing exactly one element is called singleton set or unit set. For example, A = {1}, A = {x: x=1}, etc.
3. Finite set: A set which contains a finite number of elements is called the finite set. For example, A = {1},
A = {x: x=1}, etc.
4. Infinite set: A set which contains infinite number of elements is called the infinite set. For example,
N = {1, 2, 3,4, ………}, N = {x: x is a natural number}, etc.
5. Equivalent set: A set containing same number of elements is called equivalent set. For example, let A = {1} and
B = {2} then, A and B are equivalent set.
6. Equal set: Two sets A and B are said to be equal if all the elements of A are present in B and vice-versa. For example, if, A = {1, 2, 3} and B = {x: 0 < x > 4}, then A and B are said to be equal sets.
7. Subset: A set A is said to be a subset of set B if all the elements of set A is also present in the set B. For example, let A = {1, 2} and B = {3, 2, 1} then, A is called the subset of B or A B.
8. Power set: The collection of all the subsets of a set A is called the power set of A. It is represented by P(A). For example, if A = {1, 2} then P(A) = {{1}, {2}, {1, 2}}
9. Disjoint set: If A and B are two sets such that A ꓵ B = Φ , then A and B are called disjoint sets.
10. Universal set: A set containing all objects or element and of which all other sets are subset is called universal set. For example, if a set U = {1, 2, 3, ………} and A is another set such that A = {1,2,3} then, the set U is called the universal set. Universal set is represented by U.
SOME IMPORTANT CHARACTERISTICS OF SETS
1. Elements, objects and members of a set are synonymous terms.
2. Sets are represented by capital letters of the English alphabets.
3. Elements of a set are represented by small letters of the English alphabets.
4. Elements are not generally repeated in a set.
5. The order in which elements occurs does not make a set different.
6. All infinite sets cannot be represented in the roster form.
IMPORTANT SYMBOLS USED
SYMBOLS
|
NAME
|
READ AS
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ϵ
|
Epsilon
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Belongs to
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ꓵ
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Intersection
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Intersection
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∪
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Union
|
Union
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{ }
|
Empty set
|
Empty set
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Φ
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Phi
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Empty set
|
⊂
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Subset
|
Subset
|
OTHER IMPORTANT TERMS
1. The union of sets A and B is the set C which consists of all the sets which are either present in A or B including those which are available in both.
2. The intersection of sets A and B is the set of all the elements which are common to both A and B.
3. The relationships, between sets can be represented by the means of diagram which is known as Venn diagram.
4. The differences of the set A and B in this order is the set of elements which belongs to A but not to B.
5. Interval is a set of real numbers between two given integers or numbers.
6. Let U be the universal set and A, a subset of U, then, the compliment of A is the set of all elements of U which are not the element of A.
PROPERTIES OF THE OPERATION OF UNION
1. A ∪ B = B ∪ A (Communicative law).
2. (A ∪ B) ∪ C = A ∪ (B ∪ C) (Associative law).
3. A ∪ Φ = A (Law of identity element).
4. A ∪ A = A (Idempotent law).
5. U ∪ A = U (Law of U).
PROPERTIES OF OPERATION ON INTERSECTION
1. A ꓵ B = B ꓵ A (Communicative law).
2. (A ꓵ B) ꓵ C= A ꓵ (B ꓵ C) (Associative law).
3. Φ ꓵ A = Φ (law of Φ).
4. U ꓵ A = A (law of U).
5. A ꓵ A = A (Idempotent law).
6. A ꓵ (B ∪ C) = (A ꓵ B) ∪ (A ꓵ C) (Distributive law).
TYPES OF INTERVALS
1. Open intervals: the interval which does not contains the end points is called open interval.
· For example, let a set of real numbers, R = {y: a < y < b}. This will be denoted by (a, b).
· So, (a, b) = {y: a < y < b}.
2. Closed intervals: the interval which contains the end points is called closed interval.
· For example, let a set of real numbers, R = {x: a ≤ x ≤ b}. This will be denoted by [a, b].
· So, [a, b] = {x: a ≤ x ≤ b}.
For example, let a set of real numbers, R = {y: a ≤ x < b}. This will be denoted by [a, b). This is a set which include a but does not include b.
· So, [a, b) = {y: a ≤ x < b} and this will be read as ‘an open interval from a to b including a but excluding b.
Similarly, let a set of real numbers, R = {y: a < x ≤ b}. This will be denoted by (a, b]. This is a set which include b but does not include a.
· So, (a, b] = {y: a < x ≤ b} and this will be read as ‘an open interval from a to b including b but excluding a.
PROPERTIES OF COMPLEMENT SETS
1. A ∪ AI = U, A ꓵ AI = Φ (Complement laws)
2. (A ∪ B) = A B, (A B) = A ∪ B (De Morgan’s law)
3. (AI)I = A (law of double complementation)
4. ΦI = U (law of empty set)
5. UI = Φ (law of universal set)
IMPORTANT MEANINGS TO BE UNDERSTAND
1. Let A and B be two sets such that,
· A = {1, 2, 3}, B = {1, 2, 3, 4}
· Then,
· A ∪ B = {1, 2, 3, 4} (A ∪ b will include all the elements present in A and B, the common elements being taken once).
· A ꓵ B = {1, 2, 3} (A ꓵ B will include only the elements present in both the sets).
2. Let A and B be two sets such that,
· A = {1, 2, 3}, B = {3, 4, 5, 6, 7}
· Then,
· A – B = {1, 2} (A – B will only include the elements that are present in only A and not in B)
3. Let U and A be two sets such that,
· U = {1, 2, 3, …., 10}, A = {1, 2, 3}
· AI = {4, 5, 6, 7, 8, 9, 10} (AI will include all the elements that are not present in set A but are present in the universal set U)