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.

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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.

Q2. Show that the set G = {x + y √2 |x,y ∊ Q} is a group with respect to addition. 

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. 

Q6. Prove that the intersection of two subgroups of a group is also subgroup. 

Ans. 

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