Definition of Binary Operations
A binary operation is a mathematical operation that combines two elements (called operands) to produce another element. It is denoted by a symbol (like *, +, ×, etc.) and must satisfy the closure property.
Formally, a binary operation * on a set S is a function:
\[ * : S \times S \rightarrow S \]For any two elements \( a, b \in S \), the operation produces \( a * b \in S \).
Common Binary Operations:
- Addition (+) on real numbers
- Multiplication (×) on integers
- Subtraction (-) on rational numbers
Notation:
We often use symbols like *, ⊗, ⊕, or ○ to represent general binary operations.
Example 1:
Let * be defined on ℤ by \( a * b = a + b - ab \). Find \( 3 * 5 \).
Closure Property
A set S is closed under a binary operation * if for every \( a, b \in S \), the result \( a * b \) is also in S.
Example 2:
Is the set of even integers closed under addition?
Yes, because the sum of any two even integers is always even.
\[ \text{Let } a = 2k, b = 2m \text{ where } k,m \in \mathbb{Z} \] \[ a + b = 2k + 2m = 2(k + m) \text{ (even)} \]Example 3:
Is the set of odd integers closed under multiplication?
Yes, because the product of any two odd integers is always odd.
\[ \text{Let } a = 2k + 1, b = 2m + 1 \] \[ a \times b = (2k + 1)(2m + 1) = 4km + 2k + 2m + 1 = 2(2km + k + m) + 1 \text{ (odd)} \]Example 4:
Is the set \( S = \{1, 2, 3\} \) closed under addition?
No, because \( 2 + 3 = 5 \) and \( 5 \notin S \).
Properties of Binary Operations
Commutative Property
A binary operation * is commutative if:
\[ a * b = b * a \quad \forall a,b \in S \]Example: Addition is commutative:
\[ 3 + 5 = 5 + 3 \]Associative Property
A binary operation * is associative if:
\[ (a * b) * c = a * (b * c) \quad \forall a,b,c \in S \]Example: Multiplication is associative:
\[ (2 \times 3) \times 4 = 2 \times (3 \times 4) \]Distributive Property
Operation * distributes over ○ if:
\[ a * (b ○ c) = (a * b) ○ (a * c) \] \[ (a ○ b) * c = (a * c) ○ (b * c) \]Example: Multiplication distributes over addition:
\[ 2 \times (3 + 4) = (2 \times 3) + (2 \times 4) \]Testing Properties
To verify properties for a custom binary operation:
- Choose arbitrary elements from the set
- Apply the operation in different orders
- Check if the equality holds
Example 5:
Let * be defined by \( a * b = a + b + ab \). Verify if * is commutative and associative on ℝ.
Commutative:
\[ a * b = a + b + ab \] \[ b * a = b + a + ba = a + b + ab \]Since \( a * b = b * a \), * is commutative.
Associative:
\[ (a * b) * c = (a + b + ab) * c = a + b + ab + c + (a + b + ab)c \] \[ = a + b + c + ab + ac + bc + abc \] \[ a * (b * c) = a * (b + c + bc) = a + b + c + bc + a(b + c + bc) \] \[ = a + b + c + bc + ab + ac + abc \]Since both expressions are equal, * is associative.
Identity Element
An identity element (or neutral element) for a binary operation * on set S is an element \( e \in S \) such that:
\[ a * e = e * a = a \quad \forall a \in S \]Additive Identity:
For addition, the identity is 0:
\[ a + 0 = 0 + a = a \]Multiplicative Identity:
For multiplication, the identity is 1:
\[ a \times 1 = 1 \times a = a \]Example 6:
Find the identity element for * defined by \( a * b = a + b - ab \) on ℝ.
Let e be the identity:
\[ a * e = a \] \[ a + e - ae = a \] \[ e - ae = 0 \] \[ e(1 - a) = 0 \]This must hold for all a, so e = 0.
Verify: \( a * 0 = a + 0 - a \times 0 = a \) ✔️
Example 7:
Does the operation \( a * b = a + b + 1 \) on ℤ have an identity?
Let e be the identity:
\[ a * e = a \] \[ a + e + 1 = a \] \[ e + 1 = 0 \] \[ e = -1 \]Verify: \( a * -1 = a + (-1) + 1 = a \) ✔️
Also \( -1 * a = -1 + a + 1 = a \) ✔️
Thus, -1 is the identity element.
Inverse Element
Given a binary operation * on S with identity element e, the inverse of an element \( a \in S \) is an element \( a^{-1} \) such that:
\[ a * a^{-1} = a^{-1} * a = e \]Additive Inverse:
For addition, the inverse of a is -a:
\[ a + (-a) = 0 \]Multiplicative Inverse:
For multiplication, the inverse of a is 1/a (a ≠ 0):
\[ a \times \frac{1}{a} = 1 \]Example 8:
Find inverses under * defined by \( a * b = a + b - ab \) (identity is 0).
Find \( a^{-1} \) such that \( a * a^{-1} = 0 \):
\[ a + a^{-1} - a \times a^{-1} = 0 \] \[ a^{-1}(1 - a) = -a \] \[ a^{-1} = \frac{a}{a - 1} \text{ for } a \neq 1 \]Thus, the inverse is \( \frac{a}{a - 1} \).
Example 9:
For \( a * b = a + b + 1 \) (identity is -1), find the inverse of 5.
Find \( 5^{-1} \) such that \( 5 * 5^{-1} = -1 \):
\[ 5 + 5^{-1} + 1 = -1 \] \[ 5^{-1} = -1 - 5 - 1 = -7 \]Verify: \( 5 * -7 = 5 + (-7) + 1 = -1 \) ✔️
Important Notes
- Not all elements may have inverses
- An element may be its own inverse (e.g., -1 for multiplication)
- The identity element is always invertible and is its own inverse
Practice Problems
Problem 1
Define * on ℤ by \( a * b = a + b + 2 \). Show that * is commutative and associative.
Commutative:
\[ a * b = a + b + 2 = b + a + 2 = b * a \]Associative:
\[ (a * b) * c = (a + b + 2) * c = a + b + 2 + c + 2 = a + b + c + 4 \] \[ a * (b * c) = a * (b + c + 2) = a + b + c + 2 + 2 = a + b + c + 4 \]Both expressions are equal, so * is associative.
Problem 2
Let * be defined on ℝ by \( a * b = \frac{a + b}{1 + ab} \). Find the identity element and the inverse of 2.
Identity:
\[ a * e = a \Rightarrow \frac{a + e}{1 + ae} = a \] \[ a + e = a + a^2e \] \[ e - a^2e = 0 \] \[ e(1 - a^2) = 0 \]For all a ≠ ±1, e must be 0.
Verify: \( a * 0 = \frac{a + 0}{1 + a \times 0} = a \) ✔️
Inverse of 2:
\[ 2 * 2^{-1} = 0 \Rightarrow \frac{2 + 2^{-1}}{1 + 2 \times 2^{-1}} = 0 \] \[ 2 + 2^{-1} = 0 \] \[ 2^{-1} = -2 \]Verify: \( 2 * -2 = \frac{2 + (-2)}{1 + 2 \times (-2)} = \frac{0}{-3} = 0 \) ✔️
Problem 3
Determine if the operation ○ defined by \( a ○ b = |a - b| \) on ℕ is associative.
Test with specific values:
\[ (3 ○ 2) ○ 1 = |3 - 2| ○ 1 = 1 ○ 1 = |1 - 1| = 0 \] \[ 3 ○ (2 ○ 1) = 3 ○ |2 - 1| = 3 ○ 1 = |3 - 1| = 2 \]Since 0 ≠ 2, ○ is not associative.
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