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.
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.
Contents [hide]
- 1
What is a Binary Operation?
- 2
C – Closure Axiom
- 3
A – Associative Axiom
- 4
I – Identity Axiom
- 5
I – Inverse Axiom
- 6
C – Commutative Axiom
- 7
Summary Table:
- 8 Day 06Part 03- Binary Operation CAIIC Axioms Closer Associative Identity Inverse and Commutative.
- 9 Algebraic Structure Binary Operation on A Set OPERATIONS
- 10 Commutative Associative Binary Operations on a Set with …
- 11 1. The Axioms A binary operation on a set is …
- 12 section 1 binary operations, sets and binomial
- 13 Binary Operations
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
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.
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
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
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
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
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 |
If you’d like examples for multiplication, matrix operations, or modular arithmetic, I can include those too!