Day 06Part 02- Discrete Mathematics for Gate Computer – Algebraic Structure and Binary operations.
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Here’s a detailed learning guide for:
📘 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
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Set + Binary Operation
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Operation is Associative
🔹 Monoid
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Semigroup + Identity Element
🔹 Group
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Monoid + Inverse Exists for All Elements
🔹 Abelian Group
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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:
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Closure: Yes, all results in {0,1,2}\{0,1,2\}
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Associative: Yes (mod addition is associative)
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Identity: 0 (since a+0mod 3=aa + 0 \mod 3 = a)
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Inverse:
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0 ↔ 0
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1 ↔ 2
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2 ↔ 1
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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:
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Algebraic structures
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Group Theory
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Boolean Algebra
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Modular Arithmetic
Would you like:
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A PDF notes version?
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More GATE MCQs with solutions?
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Visual diagram of structure hierarchy?
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