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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.
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| Symbolically, (a,b) = (c,d) if a=c, b=d. |
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- The cartesian product of two non-empty sets A and B is defined
as the set of all ordered pairs (a, b), where a Î
A b Î B and is denoted by A x B.
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| Symbolically, |
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- Relation from a set A to a set B is a subset of A x B.
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- A relation R on a set A is reflexive if a R a,
"
a Î A |
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- A relation R on a set A is symmetric if a R b
Þ
b R a. |
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- A relation R on a set A transitive if a R b and b R c
Þ a R c. |
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- A relation R is said to be an equivalence relation on A if it is reflexive, symmetric and transitive.
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- A relation f from A to B is said to be a function if every element of A occurs as the first co-ordinate of at least one ordered-pair in f and f does not contain two ordered pairs having the same first co- ordinate.
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- Let A and B be two non-empty sets. A correspondence between the elements of A and B is called a function from A to B if to each element of A, there corresponds exactly one element of B.
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- Let f : A → B be a function,
then:
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| (a) A is called the domain of f |
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| (b) B is called the co-domain of f |
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| (c) The set {f(x) : x
Î A} is called the range of f. Range (f) is a subset of B |
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- A function f : A→ B is said to
be one-one or injective if f(x1) = f(x2)
Þ x1 = x2.
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- A function f : A→ B is said to
be onto or surjective if every bÎB is the
image if some a Î A.
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- A function f : A→ B is bijective
if f is one-one and onto.
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- A function f : X→ Y is called
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| Here p(x) and q(x) are real polynomials. |
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- (i) Every function from A to B is a relation from A to B
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| (ii) A relation from A to B is a function from A to B only when |
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| (a) each element of A is the first element of some element in R and |
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| (b) no element of A is the first element of any two distinct elements in R. |
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- A function f : A→B is called a
one-one function if the images of distinct elements of A are also
distinct elements of B
Symbolically
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| A one-one function is also called an injective function. |
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- f : A→B is not 1 - 1 if there
exists two distinct elements in A, whose images are same. If f : A→B
is not 1 - 1 then it is said to be a many - one function.
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- A function f : A→B is called an
onto function if every element of B is the image of at least one element
of A
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| An onto function is also called a subjective function. |
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- If f : A→B is a onto function,
then range (f) = B.
If A→B
is not onto, then there exists at least one element in B which is not the
image of any element in A.
If f : A→B
is not onto, then it said to be an into function. |
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- A function f : A→ B is called a
one-one, onto function if it is both one-one and onto.
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- (i) Let f : A→B and g : B→C
be two functions. The function gof : A→C
defined by (gof)(a) = g(f(a)), a ÎA is called
the composite function of f and g.
(ii) If f and g be
functions from A to A then fog and gof are both defined. In general fog
¹ gof. |
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- A function from A x A into A is called a binary operation on A.
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