Day 06Part 07- Discrete mathematics for computer engineering- Trick for finding of group of finite numbers.

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Here’s a simplified and trick-based explanation of “How to identify a group from a set of finite numbers” — perfect for Discrete Mathematics (Day 06 Part 07), especially if you’re studying for Computer Engineering or GATE.

What is a Group? (Basic Definition)

In group theory, a group is a set $G$ along with a binary operation $*$ that satisfies four properties:

Property	Description
Closure	For all $\in G$ , $\in G$
Associativity	For all $\in G$ , $(a * b) * c = a * (b * c)$
Identity	There exists an element $\in G$ such that $a * e = e * a = a$
Inverse	For every $\in G$ , there exists $a−1∈Ga^{-1} \in G$ such that $a * a^{-1} = e$

Shortcut/Trick to Check if a Set is a Group

Let’s say you’re given a finite set with a binary operation like addition or multiplication (modulo something), and you’re asked:

“Is this a group?”

Follow this Trick-Step Checklist:

Step 1: Check Closure

Try a few pairs $a, b$ and verify that $a * b$ is also in the set.

Tip: If any result is outside the set → Not a group.

Step 2: Check Associativity

Check if $(a * b) * c = a * (b * c)$

Tip: For addition and multiplication modulo n, associativity always holds.
So you can skip manual check if it’s modulo.

Step 3: Check Identity Element

Find an element $e$ such that $a * e = a$ for all $\in G$

Addition: $0$ is identity
Multiplication: $1$ is identity

Step 4: Check Inverse

For every $\in G$ , check if there exists an $a−1∈Ga^{-1} \in G$ such that $a * a^{-1} = e$

Use a table to test inverses in small sets.

Example 1: Set ${0, 1, 2\}$ under Addition Mod 3

We check:

Closure: $\mod 3 \in \{0,1,2\}$
Associative: Addition mod n is associative
Identity: $0$
Inverse:
- $0^{-1} = 0$
- $1^{-1} = 2$
- $2^{-1} = 1$

So it’s a group under addition mod 3.

Example 2: Set ${1, 2, 3\}$ under Multiplication Mod 4

Let’s test:

$\mod 4 = 0 \notin \{1,2,3\}$ Not closed

So this is NOT a group

Quick Tips:

Addition mod n over $Zn\mathbb{Z}_n$ → Always a group.
Multiplication mod n over $Zn\mathbb{Z}_n$ → Group only if elements are coprime with n.
Use group table method for small sets.

Final Trick Formula:

If a finite set $G$ with operation $*$ satisfies closure, identity, associativity, and every element has an inverse → G is a group.

Let me know if you’d like:

A PDF with solved examples
A table method to verify group properties
A video lesson explaining this with animations

Trick for Finding a Group of Finite Numbers in Discrete Mathematics

Quick Trick to Check if a Finite Set Forms a Group

Example 1: Checking if (Z₆, +) is a Group

Example 2: Checking if (Z₆, ×) is a Group

Shortcut Rule for Small Sets