LEVEL-2
15. Given the relation R = {(1, 2), (2, 3)) on the set A = (1, 2, 3), add a minimum number of
ordered pairs so that the enlarged relation is symmetric, transitive and reflexive.
16. Let A - (1,2,3) and R = {(1,2), (1, 1), (2,3)} be a relation on A. What minimum number of
ordered pairs may be added to R so that it may become a transitive relation on A.
17. Let A-la,b,d and the relation R be defined on A as follows: R=(a, a),(b, c),(a, b)). Then,
write minimum number of ordered pairs to be added in R to make it reflexive and
transitive.
8. Each of the following defines a relation on N:
() *>y, x, y eN
(ii) x + y = 10, x, y eN
(l) xyis square of an integer,x,y eN (iv) x + 4y =10, x, y eN
Determine which of the above relations are reflexive, symmetric and transitive.
INCERT EXEMPLARI
ANSWERS
Reflexive, symmetric and transitive
() Reflexive, symmetric and transitive
() Neither reflexive, nor symmetric but transitive
Answers
Answer:
15. R is reflexive if it contains (1,1)(2,2)(3,3)
∵(1,2)∈R, (2,3)∈R
∴R is symmetric if (2,1),(3,2)∈R
Now, R={(1,1),(2,2),(3,3),(2,1),(3,2),(2,3),(1,2)}
R will be transitive if (3,1);(1,3)∈R. Thus, R becomes and
equivalence relation by adding (1,1)(2,2)(3,3)(2,1)(3,2)(1,3)(1,2). Hence,
the total number of ordered pairs is 7.
16. Given, A={1,2,3} and R={(1,2),(1,1),(2,3)}.
Now this relation is not transitive as (1,2)∈R,(2,3)∈R but (1,3)
∈R.
So to make R transitive we are to add this order pair.
Then the relation will be R={(1,2),(1,1),(2,3),(1,3)}.
This relation R is transitive.
So minimum one pair is to be added to make R symmetric.
Explanation:
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