3.1 Binary Operations and Their Properties

Binary operations are the building blocks of algebraic structures. They're functions that combine two elements of a set to produce another element in that same set. Understanding their properties is crucial for grasping more complex algebraic concepts.

Key properties of binary operations include closure, associativity, commutativity, and the existence of identity and inverse elements. These properties help us classify and analyze different algebraic structures, setting the stage for deeper exploration of groups, rings, and fields.

Binary Operations and Properties

Definition and Key Properties

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• A binary operation on a set $S$ is a function that assigns to each ordered pair of elements of $S$ a unique element of $S$
• Closure: A set $S$ is closed under a binary operation if for all elements $a$ and $b$ in $S$, the result of the operation [a * b](https://www.fiveableKeyTerm:a_*_b) is also an element of $S$
  • Example: The set of integers is closed under addition since the sum of any two integers is always an integer
• Associativity: A binary operation $*$ on a set $S$ is associative if $(a * b) * c = a * (b * c)$ for all elements $a$, $b$, and $c$ in $S$
  • Example: Addition of real numbers is associative, as ([a + b](https://www.fiveableKeyTerm:a_+_b)) + c = a + (b + c) for all real numbers $a$, $b$, and $c$
• Commutativity: A binary operation $*$ on a set $S$ is commutative if $a * b = b * a$ for all elements $a$ and $b$ in $S$
  • Example: Multiplication of rational numbers is commutative, as $a \times b = b \times a$ for all rational numbers $a$ and $b$

Identity and Inverse Elements

• Identity element: An element $e$ in $S$ is an identity element for a binary operation $*$ if $a * e = e * a = a$ for all elements $a$ in $S$
  • Example: The number 0 is the identity element for addition on the set of real numbers, as $a + 0 = 0 + a = a$ for all real numbers $a$
• Inverse element: For a binary operation $*$ on a set $S$ with identity element $e$, an element $b$ in $S$ is an inverse of an element $a$ in $S$ if $a * b = b * a = e$
  • Example: In the set of non-zero real numbers under multiplication, the inverse of a number $a$ is $\frac{1}{a}$, as $a \times \frac{1}{a} = \frac{1}{a} \times a = 1$ (the identity element for multiplication)

Identifying Binary Operations

Verifying the Binary Operation Criteria

• Check if the operation assigns a unique element of the set to each ordered pair of elements from the set
  • Example: The operation of adding the coordinates of two points in the plane is a binary operation on the set of points, as it assigns a unique point to each pair of points
• Verify that the operation is well-defined, meaning that the result of the operation is independent of the representation of the elements
  • Example: The operation of multiplying two fractions is well-defined, as the result is the same regardless of how the fractions are represented (e.g., $\frac{1}{2} \times \frac{2}{3} = \frac{2}{6} = \frac{1}{3}$)
• Ensure that the operation does not produce any elements outside the given set (closure property)
  • Example: The operation of dividing two integers is not a binary operation on the set of integers, as it can produce non-integer results (e.g., $3 \div 2 = 1.5$, which is not an integer)

Using Counterexamples

• Counterexamples can be used to disprove that an operation is a binary operation
  • Example: The operation of subtracting a larger number from a smaller number is not a binary operation on the set of natural numbers, as it can produce negative results (e.g., $2 - 5 = -3$, which is not a natural number)
• A single counterexample is sufficient to show that an operation is not a binary operation on a given set
  • Example: To show that division is not a binary operation on the set of integers, the counterexample of $1 \div 2 = 0.5$ (a non-integer) is enough

Proving Binary Operation Properties

Proving Properties Using Definitions

• To prove a property, such as associativity or commutativity, use the definition of the binary operation and the elements of the set to show that the property holds for all possible cases
  • Example: To prove that addition is associative on the set of real numbers, show that $(a + b) + c = a + (b + c)$ for all real numbers $a$, $b$, and $c$ by expanding the expressions and simplifying
• To disprove a property, find a counterexample that demonstrates the property does not hold for at least one case
  • Example: To disprove that subtraction is commutative on the set of integers, provide a counterexample such as $3 - 5 \neq 5 - 3$
• Closure can be proven by showing that the result of the operation on any two elements of the set always produces an element within the set
  • Example: To prove that the set of even integers is closed under addition, show that the sum of any two even integers is always an even integer

Proving Identity and Inverse Elements

• Identity and inverse elements can be proven by finding elements that satisfy the respective definitions for all elements in the set
  • Example: To prove that 1 is the identity element for multiplication on the set of real numbers, show that $a \times 1 = 1 \times a = a$ for all real numbers $a$
• To prove the existence of inverse elements, find a formula or rule that generates the inverse of any element in the set
  • Example: To prove that every non-zero real number has a multiplicative inverse, show that for any non-zero real number $a$, the number $\frac{1}{a}$ satisfies the inverse property: $a \times \frac{1}{a} = \frac{1}{a} \times a = 1$

Constructing Binary Operations

Operations on Numerical Sets

• Create binary operations on numerical sets (integers, rational numbers, real numbers) by defining arithmetic operations or more complex functions that satisfy the desired properties
  • Example: Define a binary operation $*$ on the set of integers by $a * b = a + b - ab$. This operation is commutative and associative, with an identity element of 0
• Develop binary operations on non-numerical sets, such as sets of matrices, functions, or algebraic structures, by defining operations that exhibit the required properties
  • Example: Define a binary operation $\circ$ on the set of 2x2 matrices by $(A \circ B)_{ij} = \sum_{k=1}^{2} A_{ik}B_{kj}$. This operation is associative and has an identity element (the 2x2 identity matrix) but is not commutative

Constructing Examples and Counterexamples

• Provide examples of binary operations that lack certain properties, such as non-associative or non-commutative operations, to illustrate the distinctions between different types of binary operations
  • Example: The subtraction operation on the set of integers is not associative, as $(a - b) - c \neq a - (b - c)$ in general (e.g., $(5 - 3) - 2 \neq 5 - (3 - 2)$)
• Construct finite tables (Cayley tables) to represent binary operations on small finite sets, making it easier to verify the properties of the operations
  • Example: Create a Cayley table for the binary operation $*$ on the set $\{0, 1\}$ defined by $0 * 0 = 0$, $0 * 1 = 1$, $1 * 0 = 1$, and $1 * 1 = 0$. The table makes it easy to check that the operation is commutative and associative, with 0 as the identity element