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

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

Trick for Finding a Group of Finite Numbers in Discrete Mathematics

In Discrete Mathematics, a group is a set of finite numbers with a binary operation that satisfies the following properties:
Closure – If $a,b$ are in the set, then $a\ast b$ is also in the set.
Associativity – $(a\ast b)\ast c=a\ast (b\ast c)$ for all elements in the set.
Identity Element – There exists an element $e$ such that $a\ast e=e\ast a=a$.
Inverse Element – For each $a$, there is an element a−1a^{-1}a−1 such that a∗a−1=ea * a^{-1} = ea∗a−1=e.

 Quick Trick to Check if a Finite Set Forms a Group

Check Closure → Perform the operation on all elements and ensure the result is still in the set.
Check Associativity → Pick any three elements and verify the associative property.
Find the Identity Element → Find an element $e$ such that $a\ast e=a$ for all $a$.
Find Inverses → Each element should have an inverse under the given operation.

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

Consider the set Z₆ = {0, 1, 2, 3, 4, 5} under addition modulo 6.

Closure: Adding any two numbers from Z₆ modulo 6 gives another element in Z₆.
Associativity: Modulo addition is associative.
Identity Element: 0 is the identity because $a+0=a$.
Inverses Exist: Each element has an inverse:

• 1 → 5 (because $1+5\equiv 0\mathrm{mod}6$)

• 2 → 4 (because $2+4\equiv 0\mathrm{mod}6$)

• 3 → 3 (self-inverse)

Conclusion: (Z₆, +) is a Group.

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

Consider Z₆ = {0, 1, 2, 3, 4, 5} under multiplication modulo 6.

Closure: Some results are not in Z₆ (e.g., $2\times 3=6\equiv 0\mathrm{mod}6$, but 6 is not in Z₆).
Identity: There is no element $e$ such that $a\times e=a$ for all $a$.
Inverses: Some elements have no inverses.

Conclusion: (Z₆, ×) is NOT a Group.

 Shortcut Rule for Small Sets

 If the operation is addition modulo n, it usually forms a group.
 If the operation is multiplication modulo n, check for the existence of an identity element and inverses.
 The set of nonzero elements modulo a prime number (Zₚ{0}, × mod p) always forms a group.

Would you like more examples or specific tricks for other types of groups?

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