North Maharashtra University 2008 B.Sc Mathematics S.YBSc MTH – 212 (B) (Computational Algebra) -university question paper

c) {1, 4, 9} d) {1, 4, 9, 16}
2) Let A = {1, 2, 3, 4} , B = {1, 4, 6, 8, 9} and R be a relation from A
to B described by aRb ? b = a2. Then Ran(R) = - - - -
a) {1, 2, 3, 4} b) {1, 2, 3}
c) {1, 4, 9} d) {1, 4, 9, 16}
3) Let A = {1, 2, 3, 4, 6, 9, 12} and R be a relation on A described by
aRb ? a is a multiple of b. Then R-relative set of six is - - - -
a) {1, 2, 3, 6} b) {6, 12}
c) {1, 2, 3} d) {12}
4) A relation R on a set A is reflexive if and only if - - - -
a) all diagonal entries of MR are one and non diagonal entries
of MR are 0
b) all diagonal entries of MR are 1
c) all diagonal entries of MR are 0
1 2
4 3
4
d) all diagonal entries of MR are 0 and non diagonal entries
of MR are 1
5) A relation R on a set A is irreflexive if and only if - - - -
a) all diagonal entries of MR are one and non diagonal entries
of MR are 0
b) all diagonal entries of MR are 1
c) all diagonal entries of MR are 0
d) all diagonal entries of MR are 0 and non diagonal entries
of MR are 1
6) Let R be a relation on a set A. Then R2
M = - - - -
a) MR ?MR b) MR ?MR c) MR ?MR d) MR??MR
7) Symmetric closure of a relation R on a set A is - - - -
a) R b) R-1 c) R ? R-1 d) R nR-1.
8) Let A = {1, 2, 3, 4}. Which of the subsequent is a partition of A?
a) {{1,2} , {3}} b) {{1,2} , {3,4}}
c) {{1,2,3} , {2,3,4}} d) {{1,2} , {2,3} , {1,2} , {2,3}}
3 : ques. of four marks
1) If R and S are equivalence relations on a set A then show that the
smallest equivalence relation containing R and S is (R ? S)8.
2) If R is a relation on A = {a1, a2, - - - , an} then show that R2
M =
MR??MR.
3) Let R be a relation on a set A. Prove that R8 is a transitive closure of
R.
4) Let A be a set with n elements and R be a relation on A. Prove that R8
= R ? R2 ? - - - -? Rn.
5
5) discuss the method of finding partitions A/R, where R is an
equivalence relation on a finite set A. Let A = {1, 2, 3, 4} and R = {(1
, 1) , (1 , 2) , (2 , 1) , (2 , 2) , (3 , 4) , (4 , 3) , (3 , 3) , (4 , 4)} be an
equivalence relation on A. obtain A/R.
6) Let P be a partition of a set A. describe a relation R on A by “aRb if and

Earning:   Approval pending.