MTH405 Quiz by Abdul Hadi E Services

MTH405, MTH633 Assignment 1 Solution

Solved by: Abdul Hadi E Services

Total Marks: 10 Due Date: December 21, 2021

Q1. Find the order of each element of the group {1, -1, i, -i} Marks = 5

Solution:

Let 1, 1, , G i i Because this group under multiplication so e = By using Formula n a e a G Here n is called the order of the element. Putting a =1 and n = 1 in the formula (1) e So, 1 is order of 1 Now, Putting a = -1 and n = 2 in the formula ( 1) e So, 2 is order of -1 Similarly, Putting a = i and n = 4 in the formula ( 1) e So, 4 is order of i And, Putting a = - i − and n = 4 in the formula i e So, 4 is order of i

Q2. Show that the set of integers ‹𝐙, +› is a group under addition. Marks = 5

Solution:

Case 1:

There exists an identity element in the group that fixes every element under given the binary operation. Yes, the number 0 is the identity since k k o k Z

Case 2:

Closure: given any two-element a and b in the set, we need to show that a+ b in the set. Yes, given any two integers a and b, there same a + b is again an integer.

Case 3:

Associativity: Yes, addition in Z is associative. i.e., ( a b c a b c

Case 4:

Existent of inverse: given any element a in the set, we need to find another element such that a and its inverse commute and their operation together given the identity element. Yes, given a G then -a is the inverse. Since a a a a

Since Z with addition satisfies above all 4 conditions. So, Z +  is a group under addition