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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.

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

**Ans. **

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

_{6}

_{6}

_{6}

**Ans. **The composition table is:

4 + _{6}5 = 3

From the table we get the following observations

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

**Associativity:** The composition ‘+_{6}’ is associative. If a, b, c are any three elements of Z_{6},

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

∴(Z_{6} + _{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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