In November 2024
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F ind aba -1 w here (i) a = (5 , 7 , 9) , b = (1 , 2, 3) (ii) a = (1 , 2,5)(3 , 4) , b = (1 , 4 , 5) .
If G is the abelian group of integers in the m apping T: G → G given by T(x ) = x then prove that as an autom orphism
1. Prove: A∪B∪C=A∪(B∪C)
2. Prove: A∩B'=A'∪B'
3. Given nU=692, nA=300, nB=230, nC=370, nA∩B=150, nA∩C=180, nB∩C=90, nA∩B'∩C'=10 where n(S) is the number of distinct elements in the set S, find:
a. n(A∩B∩C) c. n(A'∩B'∩C')
b. n(A'∩B∩C') d. n((A∩B)∪(A∩C)∪(B∩C)
4. Show that total number of proper subsets of S={a1,a2,… an} is 2n=1.
5. Show that multiplication is a binary operation on S={1,-1, i, -i} where i=-1.
6. Show that “is a factor of” on N is reflexive and transitive but is not symmetric.
If H and K is subgroup prove H intersection K is subgrop.
[DMe] Define group. Show that the set P3 of all permutations on three symbols 1,2,3 is a finite non-abelian group of order six with respect to permutation multiplication as composition.
if G is the abelian group of integers in the mapping T:G to G given by T(x) = x then prove that as an automorphism
Let
nn ∈N
a )( be any sequence. Show that Lan
n
=
∞→
lim iff for every ,0 ε > there exists
some N ∈ N such that ≥ Nn implies )
If A,B C are three ideals of a ring R then show that A(BC)=(AB)C
Show that a ring can't be expressed as union of 2 proper ideals but it is possible to express it as a union of three proper ideals
P rove that all real of the form a + b 2 , a; b ∈ Z form s a ring.
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#340153 on Dec 2023