Binary Operation & Binary Structure: Standard Sets in Abstract Algebra

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  • 0:03 Binary Operations
  • 0:32 Sets
  • 1:29 Addition, Subtraction, Other
  • 2:32 Classifying Binary Operations
  • 3:45 Binary Structure
  • 5:00 Lesson Summary
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Lesson Transcript
Instructor: Michael Gundlach
In mathematics, we like to combine things together in different ways to try to figure out what they make. We use binary operations to combine things together, and understanding such operations is vital in the study of abstract algebra.

Binary Operations

Do you remember having to memorize times tables and division facts in grade school? What about learning how to solve one-, two-, and three-digit addition and subtraction equations and doing endless amounts of practice problems? The study of the four basic operations - addition, subtraction, multiplication, and division - are examples of binary operations, and the study of binary operations forms the foundation upon which abstract algebra is built. So, what is a binary operation?


In order to formally define a binary operation, we need to first talk about sets. A set is a collection of objects, where the objects are in no particular order and there are no repeats. Think of a set as a mathematical box; stuff is just thrown in the box in no particular order, but we can describe what's in it. We usually denote sets with capital letters, such as S. Throughout this lesson, we'll often use S to denote an arbitrary set. We often call the things in the box elements, which could include numbers, words, functions, or even cats, if we wanted to get really crazy.

A binary operation takes two elements of a set S and spits out a third element, also from the set S. Think of a binary operation as a mathematical machine that takes two inputs and produces one output. The inputs and outputs are always from the same set S. In general, the order of the input matters: let's look at some examples of binary operations.

Addition, Subtraction, Other

Consider the binary operations of addition and subtraction, the first ones we learn in grade school. When we use the binary operation of addition, we can take two whole numbers, like 1 and 2, and add them, like so: 1 + 2 = 3. In this example, 1 is the first input, 2 the second input, and 3 the output. Similarly, we can subtract: 2 - 1 = 1; in this example, 2 is the first input, 1 the second input, and 1 the output.

Other Binary Operations

Multiplication and division are also binary operations. If you've ever worked with matrices, matrix addition and multiplication are further examples of binary operations. Function composition is another example that is studied extensively in abstract algebra.

We can also come up with our own binary operations by combining known operations. For example, we could define a binary operation ♦ by defining ab = a ÷ 3 + 2b − 25, one of many possible operations.

Classifying Binary Operations

Since there are so many binary operations, many of which can be rather nasty looking, like the diamond we defined above, we like to classify them according to their properties, specifically commutative and associative operations. When talking about these operations, we'll use an asterisk (∗) to represent any arbitrary binary operation, just as variables are often used to represent arbitrary numbers.

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