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| Relation |
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| Consider the following sentences: |
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| 42 is a multiple of 6. |
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| Line l is parallel to line m. |
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| 10 is greater than 7. |
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| In each case the first element is related to the last element by the relation in italics. The two elements named are members of two separate sets, say set A and set B. |
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If and we obtain a set of ordered pairs (x, y). Thus a relation forms a set of ordered pairs. |
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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. |
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| Let R mean 'is greater than', then xRy means x > y. |
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| If A = {1, 2, 3} and B = {3, 5} list the ordered pairs for each of the following relations: |
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(i)  |
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(ii)  |
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(iii)  |
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| A x B = {(1, 3), (1, 5), (2, 3), (2, 5), (3, 3), (3, 5)} |
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(i)  |
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| = {(3, 3)} is the only pair which satisfies the relation x = y. |
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| R 'is equal to' is a subset of A x B. |
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(ii) S = {(x, y) : x, y A x B, y = x + 2} |
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| = {(1, 3), (3, 5)}. These two pairs satisfy the relation y = x + 2. |
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(iii)  |
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| = {(1, 3), (1, 5), (2, 3), (2, 5), (3, 5)} |
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Five pairs satisfy the relation x < y |
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R, S and T are all subsets of A x B, hence they are all relations. |
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| The set of first elements in a relation is called Domain. |
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| In example (6), Domain is set A = {1, 2, 3}. |
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| The set of second elements in a relation is called Range. |
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| In example (6), Range is set B = {3, 5}. |
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Ordered Pairs and Cartesian Product
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