Day 06Part 03- Binary Operation CAIIC Axioms Closer Associative Identity Inverse and Commutative.

Day 06Part 03- Binary Operation CAIIC Axioms Closer Associative Identity Inverse and Commutative.

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Let’s break down Day 06 Part 03: Binary Operation – CAIIC Axioms step by step. The acronym CAIIC stands for:

• Closure
• Associative
• Identity
• Inverse
• Commutative

These are axioms (rules) used to describe binary operations on a set, such as addition, multiplication, etc.

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What is a Binary Operation?

A binary operation on a set is a rule for combining any two elements of the set to form another element in the same set.

Notation: If ∗* is a binary operation on a set SS, then

∀a,b∈S,a∗b∈S\forall a, b \in S, \quad a * b \in S

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C – Closure Axiom

• The result of the operation on any two elements in the set must also be in the set.
• Example: Z\mathbb{Z} under addition → 2+3=5∈Z2 + 3 = 5 \in \mathbb{Z} → Closed.

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A – Associative Axiom

• The grouping of elements does not affect the result.

(a * b) * c = a * (b * c)
]

• Example: (2+3)+4=2+(3+4)=9(2 + 3) + 4 = 2 + (3 + 4) = 9

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I – Identity Axiom

• There exists an element e∈Se \in S such that:

a∗e=e∗a=a,∀a∈Sa * e = e * a = a, \quad \forall a \in S

• Example: For addition, 0 is the identity. a+0=aa + 0 = a

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I – Inverse Axiom

• For every element a∈Sa \in S, there exists an element b∈Sb \in S such that:

a∗b=b∗a=e(identity element)a * b = b * a = e \quad \text{(identity element)}

• Example: For addition, the inverse of 5 is -5 because 5+(−5)=05 + (-5) = 0

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C – Commutative Axiom

• The order of the elements doesn’t affect the result:

a∗b=b∗aa * b = b * a

• Example: 3+5=5+3=83 + 5 = 5 + 3 = 8

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Summary Table:

Axiom	Meaning	Example (Addition on Z\mathbb{Z})
Closure	Result is still in the set	3+4=7∈Z3 + 4 = 7 \in \mathbb{Z}
Associative	Grouping doesn’t matter	(1+2)+3=1+(2+3)(1+2)+3 = 1+(2+3)
Identity	Exists an element ee	a+0=aa + 0 = a
Inverse	Exists an element to undo op	a+(−a)=0a + (-a) = 0
Commutative	Order doesn’t matter	a+b=b+aa + b = b + a

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If you’d like examples for multiplication, matrix operations, or modular arithmetic, I can include those too!

Day 06Part 03- Binary Operation CAIIC Axioms Closer Associative Identity Inverse and Commutative.

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