12. R1 and R2 be two relations on the set of integers Z defined as below -(i) R1={(a,b):a^2=b^2 and a,b∈Z} (ii) R2={(a,b): a-b is positive where a,b are primes and both are smaller than 10}  determine the domain and the range of each of them- Advance Math Class 10- Exercise 1.3

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Related Content: relation r in the set z of all integers defined as r= (x y) x-y is an integer, if a = (1,2,3,4,5 and b=(4,5,6,7,8 find sets a∪b and a∩b also if x 1,2,3,4,5,6,7,8,9 10 find)), prove if a=b mod n then (a^k)=(b^k) mod n, find the domain and range of the relation r given by r= (x y) y=x+6/x, (ii) relation r in the set n of natural numbers defined as, the relation r defined the set a=\ 1,2,3,4,5\ by r=\ (x,y):|x^ 2 -y^ 2 |.

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Solution:

(i) 

Answer: R1 = {(1, 1), (2, 2), (3, 3),...... (1, -1), (2, -2), (3, -3),......(-1, 1), (-2, 2), (-3, 3),.... (-1, -1), (-2, -2), (3, -3),.....}

Domain, d(R1) = Z

Range, r(R1) = Z

(ii) R2={(a, b): a-b is positive where a, b are primes and both are smaller than 10}

Answer: R2 = {(3, 2), (5, 2), (3, -3), (7, 2), (5, 3),(7, 3), (7, 5)}

Domain, d(R2) = {3, 5, 7}

Range, r(R2) = {2, 3, 5}

Exercise: 1.3

1. If A = {1, 3}, then write the identity relation I : A⟶A. Also write the universal relation on A ?

2. If A = {1, 2} then write down all the relation on A ?

3. If A = (1, 2, 3) then find the elements of the relation R= {(x, y): x=y and x, y Є A} on A ?

4. Z + is the set of positive integers and R: Z+ → Z+ is a relation defined as R = {(a, b) | a, b Є Z+ and a-b >2}. Is it a finite relation? Represent R as a set in the Roster form ?

5. A={2,3,4,5} and B={3,6,7,10} be two sets and a relation R is defined as R={(x, y): x completely divides y where x Є A and y Є B}. Write the relation R in the tabular form. Also represent R by arrow diagram and matrix table ?

6. Determine R^-1 of the relation R given in question 5 above. Also find d(R^-1) and r(R^-1) ?

7. A={3,6,8,9} is a set and aRb iff a-b is divisible by 3 for a,b Є A. Then (i) Write R as a set and draw the arrow diagram. (ii) Find d(R) and r(R). (iii) Can we find the inverse of R ?

8. A={1,2,3,4} and B={a,b,c,d} be two sets. Choose which of the followings are relation from A to B ?

9. A relation R is defined on the the set of natural N as aRb where a=b^2 for a,bЄN. Write the relation R. Also write R^-1 in set builder method ?

10. If A={1,2,3,4,6} and R is a relation on A defined as R={(x,y):y is exactly divisible by x where x,yЄA} then ?

11. Let A = {10, 11, 12, 13} and B = {1, 2, 3, 4} be two sets. Write the following relations from A to B as defined in each case ?

12. R1 and R2 be two relations on the set of integers Z defined as below -(i) R1={(a,b):a^2=b^2 and a,b∈Z} (ii) R2={(a,b): a-b is positive where a,b are primes and both are smaller than 10} determine the domain and the range of each of them ?