Department of Mathematical Sciences, IIT(BHU)
Even Semester 2023-2024
MA102 - Engineering Mathematics- II
Tutorial-(1)
Q (1) : Check whether ~ is an equivalence relation on set X.
(a) X = โโโ, ~ is the relation on X defined by (a,b) ~ (c,d) โ ad(b + c) = bc(a + d).
(b) X = โค , ~ is the relation on X defined by x ~ y โ ( x โ y ) is divisible by n.
Q (2) : Define a relation ~ in the set of integers โค as follows a~b iff a+b is an even integer. Is ~
(a) Reflexive, (b) Symmetric, (c) Transitive ?
If so, write down the quotient set โค/~ .
Q (3) Test whether the given algebraic structure forms a group with respect to given operations.
If it is not, write the property which fails.
(i) (โ , +) ,( โโช {0} , +) and (โค , +) .
(ii) (โ - {-1}, ๐), where ๐ is defined by ab = a + b +ab.
(iii) ( โ โ {0} ,/) with a/b = ๏
๏ , where a, b are non zero element of Q.
(iv) ( โคโโค , โ), where โ is defined by (a,b) โ (c,d) = (ad+ bc, bd).
(v) (โ-{0}, ร) and ( โ , +)
(vi) ( โค6, +๏บ) and ( โค6\{0}, ร๏บ ).
Q (4) Test whether H is a subgroup of a group G or not (operations in G are usual addition).
(i) G = โ (a) H = โค (b) H = โ (c) H = { 0 } (d) H = โ.
(ii) G = ๓ฐฅ๓ฐฃ๐ ๐
๐ ๐๓ฐค โถ ๐, ๐, ๐, ๐ โ โ๓ฐฆ (a) H = ๓ฐฅ๓ฐฃ๐ ๐
๐ ๐๓ฐค โถ ๐ โ โ๓ฐฆ .
Q (5) Examine whether the following sets form a field or not.
(i) โ = {a + bi, a, b โ โ}
(ii) (ii) โค8
(iii) (iii) โค7 .
