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SETS AND FUNCTIONS
Sets and Their Representation:
A set is a collection of distinct elements or objects
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For example, A = {1, 2, 3, 4}
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For example, A = {x | x is an even number} represents the set of even numbers
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For example, if A = {1, 2, 3}, then |A| = 3
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Universal Set: The universal set, denoted as U, is the set that contains all the elements under
consideration
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Operations on Sets:
Union of Sets: The union of two sets A and B, denoted as A ∪ B, is the set that contains all the elements
present in A or B or both
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Example: A = {1, 2, 3}, B = {3, 4, 5}
A ∩ B = {3}
Complement of a Set: The complement of a set A with respect to a universal set U, denoted as A’, is the
set that contains all elements in U but not in A
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Example: A = {1, 2, 3}, B = {2, 3, 4}
A – B = {1}
Symmetric Difference of Sets: The symmetric difference of two sets A and B, denoted as A △ B, is the set
that contains elements present in either A or B, but not in both
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Example: Let A = {1, 2}, B = {2, 3}, C = {3, 4}
- Commutative Law: A ∪ B = B ∪ A, A ∩ B = B ∩ A
- Associative Law: (A ∪ B) ∪ C = A ∪ (B ∪ C), (A ∩ B) ∩ C = A ∩ (B ∩ C)
- Distributive Law: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C), A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

Power Set:
The power set of a set A, denoted as P(A), is the set that contains all possible subsets of A, including the
empty set and the set itself
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This can be derived from the
fact that each element in the set can either be included or excluded in each subset
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Types of Relations:
a) Reflexive Relation: A relation R on a set A is reflexive if (a, a) ∈ R for all a ∈ A
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b) Symmetric Relation: A relation R on a set A is symmetric if (a, b) ∈ R implies (b, a) ∈ R for all a, b
∈ A
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c) Transitive Relation: A relation R on a set A is transitive if (a, b) ∈ R and (b, c) ∈ R imply (a, c) ∈ R
for all a, b, c ∈ A
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Equivalence Relations:
An equivalence relation is a relation that is reflexive, symmetric, and transitive
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Partition: Equivalence classes form a partition of the set, meaning that each element belongs to exactly
one equivalence class
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The equivalence classes are [{1, 2}, {3}]
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Functions:
A function is a relation between two sets A and B, where each element in A is mapped to a unique
element in B
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Example: Let A = {1, 2, 3}, B = {4, 5, 6}, f = {(1, 4), (2, 5), (3, 6)}
F is a one-to-one function because each element in A is mapped to a unique element in B
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Example: Let A = {1, 2, 3}, B = {4, 5, 6}, f = {(1, 4), (2, 5), (3, 6)}
F is an onto function because every element in B has a pre-image in A
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Example: Let A = {1, 2, 3}, B = {4, 5, 6}, f = {(1, 4), (2, 5), (3, 6)}

F is a bijective function because it is both one-to-one and onto
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Example: Let A = {1, 2, 3}, B = {4, 5}, C = {6, 7}, f = {(1, 4), (2, 4), (3, 5)}, g = {(4, 6), (4, 7)}
(g∘f)(1) = g(f(1)) = g(4) = 6
These comprehensive study notes cover sets, their representation, set operations, power set, relations,
equivalence relations, functions, and their properties
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