August
1999 QUESTION 4 Total Marks: 20 Marks |
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| (a) | Define the function f : Z ® Z by the rule f (n) = 2n, for all integers n. (i) What property must be true of a one-to-one function? (ii) What property must be true of an onto function? (iii) Is f one-to-one? Prove or give a counter example. (iv) If f onto? Prove or give a counter example.
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[1] [1] [2] [2] |
| (b) | Let S be the set {a,b}, and define S4 to be the set of all strings over S of length 4. | |
| (i) List all sixteen elements of S4. The relation R is defined on S4 as follows: for all s, t Î S4, s Rt Û s has the same first two characters as t. (ii) Is aabb R aabb? (iii) Is abba R baab? (iv) Write down all the elements which are related to abba.
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[3]
[1] [1] [2] |
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| (c) | (i) What properties must be an equivalence
relation hold? (ii) D is the binary relation defined on R as follows: " x,y Î R, xDy Û xy ³ 0Is D an equivalence relation? Justify your answer.
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[4] [3] |