Practice Abstract Algebra Solutions
Algebra deals with operations on set of numbers or elements
. Algebra replaces number by symbols usually letters of the alphabet. Algebra follows the same rules of arithmetic. Abstract algebra is deal with advanced topics of algebra that deals with algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras, logic and foundations, counting, elementary number theory, informal set theory, linear algebra, and the theory of linear operators.
Practice Abstract Algebra Solutions - Examples
Example abstract algebra problems with solutions
Example 1: A village has 1000 men and 980 women. What part of the village are women?
Solution:
Total number of men is 1000
Total number of women is 980
Total number of persons in village = sum of men and women
1000 + 980 = 1980
'(980)/(1980)'
Answer: 5 / 9 of the village are women
Example 2: Let u be a root of the polynomial a3+3a+3. In Q(u), express (7-2u+u2)-1 in the form x+yu+zu2.
Solution:
Dividing a3+3a+3 yb a2-2a+7 gives the quotient a+2 and remainder -11.
Thus u3+3u+3 = (u+2)(u2-2u+7) - 11,
and so (7-2u+u2)-1 = (2+u)/11
= (2/11) + (1/11)u.
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Example 3: Find all ring homomorphism from Z120 into Z42.
Solution:
Let : Z114 -> Z36 be a ring homomorphism. The additive order of (1) must be a divisor of gcd(114,36) = 6, so it must belong to the subgroup 6Z36 = { 0, 6, 12, 18, 24, 30 }.
Furthermore, (1) must be idempotent, and it can be checked that in 6Z36, only 0, 6, 18, 24 are idempotent.
If (1) = 6, then the image is 6Z36 and the kernel is 6Z114. If (1) = 18, then the image is 18Z36 and the kernel is 2Z114. If (1) = 24, then the image is 12Z36 and the kernel is 3Z120.
Example 4: Prove that if (xy)2 = x2y2, then xy = yx.
Solution:
(xy)2 = xyxy = x2y2.
Hence, x-1(xyxy)y-1 = x-1(x2y2)y-1.
Thus, (x-1x)yx(yy-1) = (x-1x)xy(yy-1).
Since x-1x = yy-1 =e, we have yx = xy.
Practice Abstract Algebra Solutions - Practice
Find the solutions for practice abstract algebra problems below
Practice 1: A class has 35 girls and 40 boys. What part of the class are boys?
Answer: 8 / 15 of the class are boys
Practice 2: Find the largest possible order of an element in the group Z180 of units of the ring Z180.
Answer: 12
Modern algebra deals with binary operations. A binary operation * on a set is a rule which assigns to each ordered pair of elements to the set a unique element of the set.
We can map a binary operation on a non-empty set from S X S into S.
The image of the ordered pair (a,b) under the map * is written as a * b
a * b S for all a,b S, and S is closed under the operation *
The following are the notations of modern algebra.
# Z is the set of all integers.
# Z+ or N is the set of all positive integers.
# Q is the set of all rational numbers.
# Q* is the set of all non-zero rational numbers.
# Q+ is the set of all positive rational numbers.
# R is the set of all real numbers.
# R* is the set of all non-zero real numbers.
# R+ is the set of all positive real numbers.
# C is the set of all complex numbers
# C* is the set of all non-zero complex numbers.
We can practice the modern algebra problems using the above notations.
Example Problems for Practice on Modern Algebra:
Let us do a couple of problems on modern algebra.
# Let N be the set of all natural numbers. Let * usual be addition. Verify whether closure, associative, commutative , existence of identity , inverse satisfy.
Solution:-
Step 1 : N = set of all natural numbers.
Step 2 : * = usual addition, +
Step 3 : Let a N, b N, then a + b N => closure property is satisfied.
Step 4 : Let c N. Then a + ( b + c) = (a + b) + c for all a,b,c N => Associative property is satisfied.
Step 5 : ( a + b) = (b + a) for all a,b N => Commutative property is obeyed.
Step 6 : Identity is 0 but does not exist in N
Step 7 : Let us take a number 5 N, then its inverse is -5 which does not belong to N. Hence inverse does not exist.
Answer : The * operation satisfied closure, associative and commutative property. It does not satisfy identity and
inverse property.
Practice Algebra Modern-binary Operation on Complex Number:-
Let us do a problem on complex number.
# Show that the set G = { 1, , 2} of all cube roots of unity for an abelian group with respect to multiplication as binary operation.
Solution:-
Step 1 : G = { 1, , 2}
Step 2 : All the entries are the elements of the group. Closure property is satisfied.
_______________________
X 1 2 3 = 1 and 4 = 3() = 1 times =
_______________________
1 1 2
_______________________
2 1
_______________________
2 2 1
___________________________
Associative : Complex numbers under multiplication is associative.
Identity : 1 is the identity element of and 1 G
Inverse : 1 is the inverse of +1
is the inverse of 2
2 is the inverse of
It is an abelian group.
by: mathqa
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