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Introduction |
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In a Rectangular Cartesian System of coordinates the position of a point in a plane is determined by an ordered pair whose elements give the distances from the intersecting straight lines at right angles to each other. These lines are called the axes. The vertical line is called the Y-axis and the horizontal line is called X-axis. The meeting point O of the axes is called the origin. The distance measured along the X-axis is the x-coordinate or the abscissa. The distance measured along the Y-axis is called the y-coordinate or the ordinate. |
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Ordered Pair |
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An ordered pair has a pair of elements which occur in a definite order. |
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Take two numbers 3 and 5. If they form a pair (3,5) and the order of the numbers cannot be changed, these numbers are said to form an ordered pair. (3, 5)  (5, 3) |
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If two ordered pairs (a, b) and (x, y) are equal,  (a, b) = (x, y) then a = x, b = y. |
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Cartesian Product |
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Given two sets A and B. Then, all possible ordered pairs (x, y) obtained such that  and  is called the Cartesian product of the sets A and B. |
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Relations |
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If A and B are two non-empty sets, then a relation R in A x B is a subset of A x B. If we use the letter R to denote a relation, then we can write the relation between a and b as aRb  a is related to b.
The set of first elements in a relation is called Domain.
The set of second elements in a relation is called Range. |
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Representation of a Relation |
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Relations are Represented by:
(i) Roster form
(ii) Set builder form
(iii) By tables
(iv) Arrow diagram
(v) By graphs |
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Types of Relations |
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There are four types of relations as follows:
(i) one - one
(ii) many - one
(iii) one - many
(iv) many - many |
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Properties of Relations |
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Reflexive property: A relation R in a set A is said to be reflexive if every element of the set A is related to itself. This is if aRa where a
Î A, or (a, a) Î R for each a
Î A. |
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Transitive property: A relation R, is said to be transitive if a 'is related to' b and b 'is related to' c then a 'is related to' c,  (a, b)  R and   if aRb, bRc then aRc. |
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Symmetric property: A relation R, on a set A, is symmetric if aRb then bRa  if a 'is related to' b, b 'is related to' a for a, b
Î R. Also if (a, b)
Î R then (b, a) Î R. |
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Equivalence Relation: If a relation is (i) reflexive, (ii) transitive and, (iii) symmetric, then it is called an equivalence relation. |
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Summary of Relation properties |
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(i) R is reflexive if aRa (ii) R is transitive if aRb and bRc implies aRc
(iii) R is symmetric if aRb implies bRa. A relation which is reflexive, transitive, and symmetric is called an equivalence relation.
When any one property is not satisfied we say that the relation is not an equivalence relation. |
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Summary |
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An ordered pair has a pair of elements which occur in a definite order.
The four types of relation are: one-one, many-one, one-many and many-many. Each of these relations is either an 'onto' relation or an 'into' relation.
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