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Introduction |
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In mathematics, we come across relations such as
A is a subset of B
line l is parallel to line m
number m is less than number n.
In all these, we notice that a relation involves pairs of objects in certain order. This chapter deals with the study of relations and functions in mathematics. |
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Ordered pairs and Cartesian products |
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An ordered pair is a pair of entries in the specific order. The two entries are separated by comma and enclosed within brackets.
Equality of two ordered pairs (a,b) and (c,d).
(a, b) = (c, d), if a=c and b=d.
Let A and B be two non-empty sets, then the cartesian product of A and B denoted by
A x B = {(a, b) | aÎA, bÎB}. |
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Some Important Theorem |
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Relations |
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A relation R is a non-empty sub-set of a cartesian product. |
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A relation is a set of ordered pairs, i.e., R Ì A
x B where A and B are two non-empty sets. |
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Domain of a relation is the set of all first components. |
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Range of a relation is the set of all second components. |
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Types of Relations |
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Following are the types of relations:
Reflexive Relation, Symmetric Relation, Transitive Relation, Equivalence Relation.
A relation which is reflexive, symmetric and transitive is called the Equivalence relation. |
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On the universal set of sets, the relation "subset of" is an
antisymmetric relation.
On the set of integers, the relation "congruence" is an equivalence relation. |
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Functions (or Mappings) |
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Let A and B be two non-empty sets.
A function f from A to B is an association of every element of A to an
unique element in B.
We write this as f : A g B.
A is called the domain.
B is called the co-domain.
The following are the Types of functions (Mappings):
One-one function (or an injective map), Many-one function, Onto function (or surjective map), Bijection, Constant function, Identity function, Equality of two functions, Composite function, Inverse of an element, Inverse function. |
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Composition of two functions or Product of two functions |
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Let f:AgB and g:AgB be two functions. Thus the composition of two functions f and g denoted by gof or fog is the function from A into C defined by gof = {(a,b) for some c
Î B, (a,c) Î f and (c,b)
Î g}. |
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Binary Operations |
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binary operation: Let S be any non-empty set. An operation * is called a binary
operation on S if " a, b Î
S a * b Î S |
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Commutative law: Let * be a binary operation on the set S.
* is said to be associative in S if " a, b
Î S a * b = b * a |
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Associative law: * is said to be associative in S,
If " a, b, c
Î S a * (b * c) = (a * b) * c. |
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Summary |
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A pair of objects, written in a specified order is called an ordered pair. |
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For the ordered pair (a,b), a is called the first element and b, the second element. |
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The ordered pairs (a,b) and (c,d) are equal if their corresponding elements are equal.
Symbolically, (a,b) = (c,d) if a=c, b=d.
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Welcome! 
