Binary operation
Binary Operation
A binary operation is a calculation that combines two elements (called operands) to produce another element. More formally, a binary operation on a set is a function that takes two elements from the set and returns a single element from the set.
Definition
A binary operation on a set $S$ is a function:
$$ *: S \times S \rightarrow S $$
This means that for every pair of elements $a, b \in S$, the operation $*$ assigns a unique element $a * b \in S$.
Properties of Binary Operations
Binary operations can have several important properties. Here are some of the key properties:
- Commutative: A binary operation $*$ on a set $S$ is commutative if $a * b = b * a$ for all $a, b \in S$.
- Associative: A binary operation $*$ on a set $S$ is associative if $(a * b) * c = a * (b * c)$ for all $a, b, c \in S$.
- Identity Element: An element $e \in S$ is called an identity element for a binary operation $*$ on $S$ if $a * e = e * a = a$ for all $a \in S$.
- Inverse Element: For each element $a \in S$, there exists an element $b \in S$ such that $a * b = b * a = e$, where $e$ is the identity element. The element $b$ is called the inverse of $a$.
Examples of Binary Operations
Here are some common examples of binary operations:
- Addition (+) on the set of real numbers $\mathbb{R}$.
- Multiplication (×) on the set of real numbers $\mathbb{R}$.
- Union (∪) on the set of all subsets of a given set.
- Intersection (∩) on the set of all subsets of a given set.
Table of Differences and Important Points
| Property | Description | Example in $\mathbb{R}$ | Example in Set Theory |
|---|---|---|---|
| Commutative | Order of operands does not change the result. | $a + b = b + a$ | $A \cup B = B \cup A$ |
| Associative | Grouping of operands does not change the result. | $(a + b) + c = a + (b + c)$ | $(A \cup B) \cup C = A \cup (B \cup C)$ |
| Identity Element | There exists an element that does not change other elements when operated. | $a + 0 = a$ | $A \cup \emptyset = A$ |
| Inverse Element | For each element, there exists another that combines to the identity element. | $a + (-a) = 0$ | $A \cup A^c = U$ (where $A^c$ is the complement of $A$ and $U$ is the universal set) |
Formulas
Here are some formulas related to binary operations:
- Commutative Law: $a * b = b * a$
- Associative Law: $(a * b) * c = a * (b * c)$
- Identity Law: $a * e = e * a = a$
- Inverse Law: $a * a^{-1} = a^{-1} * a = e$
Examples to Explain Important Points
Commutativity
Addition of real numbers is commutative because for any $a, b \in \mathbb{R}$, we have $a + b = b + a$.
Example: 3 + 5 = 5 + 3 = 8
Associativity
Multiplication of real numbers is associative because for any $a, b, c \in \mathbb{R}$, we have $(a \times b) \times c = a \times (b \times c)$.
Example: (2 × 3) × 4 = 6 × 4 = 24
2 × (3 × 4) = 2 × 12 = 24
Identity Element
The number 0 is the identity element for addition because for any $a \in \mathbb{R}$, we have $a + 0 = 0 + a = a$.
Example: 7 + 0 = 0 + 7 = 7
Inverse Element
For any real number $a$, the number $-a$ is its additive inverse because $a + (-a) = 0$.
Example: 5 + (-5) = 0
In conclusion, understanding binary operations is fundamental in algebra and other areas of mathematics. These operations are the building blocks for more complex structures and concepts in mathematics.