Day 06Part 02- Discrete Mathematics for Gate Computer – Algebraic Structure and Binary operations.
Day 06Part 02- Discrete Mathematics for Gate Computer – Algebraic Structure and Binary operations.
Here’s a detailed learning guide for:
Contents
- 1 📘 Day 06 Part 02 – Discrete Mathematics for GATE CSE
- 1.1 🔢 Topic: Algebraic Structures & Binary Operations
- 1.2 ✅ 1. What is a Binary Operation?
- 1.3 ✴️ 2. Examples of Binary Operations
- 1.4 🧱 3. Algebraic Structures
- 1.5 🔁 4. Important Properties of Binary Operations
- 1.6 📝 GATE-Style Sample Question:
- 1.7 📊 Quick Revision Table:
- 1.8 📚 Useful for GATE Topics:
- 1.9 Day 06Part 02- Discrete Mathematics for Gate Computer – Algebraic Structure and Binary operations.
- 1.10 Discrete Mathematics for Computer Science
- 1.11 Discrete Mathematical Structures
- 1.12 Mathematics (Discrete Structure).pdf
📘 Day 06 Part 02 – Discrete Mathematics for GATE CSE
🔢 Topic: Algebraic Structures & Binary Operations
✅ 1. What is a Binary Operation?
A binary operation on a set S is a function:
∗:S×S→S*: S \times S \rightarrow S
That means:
For any two elements a,b∈Sa, b \in S, the result a∗b∈Sa * b \in S.
✴️ 2. Examples of Binary Operations
Operation | Set | Binary? | Reason |
---|---|---|---|
a+ba + b | Integers | ✅ Yes | Result is an integer |
a−ba – b | Natural Numbers | ❌ No | Result may be negative |
a⋅bmod na \cdot b \mod n | Zn\mathbb{Z}_n | ✅ Yes | Remains in Zn\mathbb{Z}_n |
🧱 3. Algebraic Structures
An algebraic structure is a set equipped with one or more binary operations. Here are common types:
🔹 Semigroup
-
Set + Binary Operation
-
Operation is Associative
🔹 Monoid
-
Semigroup + Identity Element
🔹 Group
-
Monoid + Inverse Exists for All Elements
🔹 Abelian Group
-
Group + Commutativity
🔁 4. Important Properties of Binary Operations
Property | Definition |
---|---|
Closure | ∀a,b∈S,a∗b∈S\forall a,b \in S, a * b \in S |
Associativity | a∗(b∗c)=(a∗b)∗ca * (b * c) = (a * b) * c |
Commutativity | a∗b=b∗aa * b = b * a |
Identity | a∗e=aa * e = a and e∗a=ae * a = a |
Inverse | For each aa, a∗a−1=ea * a^{-1} = e |
📝 GATE-Style Sample Question:
Let ∗* be a binary operation defined on the set S={0,1,2}S = \{0, 1, 2\} as
a∗b=(a+b)mod 3a * b = (a + b) \mod 3.
Is (S,∗)(S, *) a group?
✅ Solution:
-
Closure: Yes, all results in {0,1,2}\{0,1,2\}
-
Associative: Yes (mod addition is associative)
-
Identity: 0 (since a+0mod 3=aa + 0 \mod 3 = a)
-
Inverse:
-
0 ↔ 0
-
1 ↔ 2
-
2 ↔ 1
-
-
Commutative: Yes (Abelian group)
Answer: Yes, it’s an Abelian Group
📊 Quick Revision Table:
Structure | Associative | Identity | Inverse | Commutative |
---|---|---|---|---|
Semigroup | ✅ | ❌ | ❌ | ❌ |
Monoid | ✅ | ✅ | ❌ | ❌ |
Group | ✅ | ✅ | ✅ | ❌ |
Abelian Group | ✅ | ✅ | ✅ | ✅ |
📚 Useful for GATE Topics:
-
Algebraic structures
-
Group Theory
-
Boolean Algebra
-
Modular Arithmetic
Would you like:
-
A PDF notes version?
-
More GATE MCQs with solutions?
-
Visual diagram of structure hierarchy?
Just let me know!