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Homework Statement
How many different commutative binary operations can be defined on a set of 2 elements? On a set of 3 elements?
The Attempt at a Solution
I do not understand the question. Seems like an infinite number.
Dick said:If '*' is the operation and {a,b} is the set of two elements, then to define the operation you need to define a*a, a*b, b*a and b*b in the set {a,b}. That's hardly an infinite number of possibilities.
A binary operation is a mathematical operation that takes two operands and produces a single output. It is denoted by a symbol between the two operands, such as addition (+) or multiplication (x).
The number of binary operations depends on the set of elements being operated on. For example, the set of integers has a different number of binary operations than the set of real numbers. In general, there are infinitely many binary operations for any given set.
A binary operation takes two operands, while a unary operation takes only one operand. In other words, a binary operation is a two-input function, while a unary operation is a one-input function.
No, not all operations can be considered binary operations. A binary operation must be well-defined, meaning that it produces a unique output for any given pair of inputs. Additionally, it must be closed, meaning that the result of the operation belongs to the same set as the operands.
Binary operations are used in many different fields of science, including mathematics, computer science, and physics. They are used to model relationships between two elements or quantities and are essential in solving equations and analyzing data.