# UNIT – 2 : ALGEBRAIC STRUCTURES in Discrete structure Btech AKTU

Unit 2 of “ALGEBRAIC STRUCTURES” in Btech Discrete Structure AKTU investigates groups, rings, and fields, as well as their characteristics and applications.

```Dudes 🤔.. You want more useful details regarding this subject. Please keep in mind this as well.

Important Questions For Discrete Structures and Theory of Logics:
*Unit-01     *Unit-02
*Unit-03    *Unit-04
*Unit-05    *Short-Q/Ans
*Question-Paper with solution 21-22 ```

## Q1. What is algebraic structure ? List properties of algebraic system.

Ans. Algebraic structure: A non-empty set G with one or more binary operations is an algebraic structure. Assume that the binary operation * on G. If so, the structure (G, *) is algebraic.

Ans.

## Q3. Prove that (Z6, (+6)) is an abelian group of order 6, where Z6={0,1,2, 3, 4, 5}.

Ans. The composition table is:

4 + 65 = 3

From the table we get the following observations

Closure: Since all the entries in the table belong to the given set Z6. Therefore, Z6 is closed with respect to addition modulo 6.

Associativity: The composition ‘+6’ is associative. If a, b, c are any three elements of Z6,

Commutative: The composition is commutative as the elements are symmetrically arranged about the main diagonal. The number of elements in the set Z6 is 6.

∴(Z6 + 6) is a finite abelian group of order 6.

Q4. Let G = {1, – 1, i, – i} with the binary operation multiplication be an algebraic structure, where i = √-1. Determine whether G is an abelian or not.

Ans. The composition table of G is

1. Closure property: Since all the entries of the composition table are the elements of the given set, the set G is closed under multiplication.

2. Associativity: Complex numbers make up the components of G, and we are aware that their multiplication is associative.

3. Identity: Here, 1 is the identity element.

4. Inverse: From the composition table, we see that the inverse elements of 1,-1, i,-i are 1,-1,-i, i respectively.

5. Commutativity: The table’s related rows and columns have the same contents. The binary operation is hence commutative. As a result, the group (G, *) is abelian.

## Q5. Let G be a group and let a, b ∊ G be any elements.

Then

i. (a-1)-1 = a

ii. (a * b)-1 * a-1.

Ans.

Ans.