What are Binary Operations? | Abstract Algebra

What are binary operations? Binary operations are a vital part of the study of abstract algebra, and we'll be introducing them with examples and proofs in this video lesson!

A binary operation on a set S is simply a function f from SxS to S. So a binary operation is a function that takes two elements from the same set and maps that pair to exactly one element also in the same set.

For example, consider addition on the real numbers. Addition takes each pair of real numbers (an element of RxR) and maps it to exactly one element also in R. So if f: RxR to R is defined by addition, then f((3,2)) = 3 + 2 = 5. This is a binary operation. Since any two real numbers add to another real number, we say the real numbers are closed under addition.

Also, addition on the real numbers is commutative, as in the order doesn't matter. For any two real numbers a and b, a + b = b + a. Some binary operations have this property and some do not.

More Examples of Binary Operations:    • Examples of Binary Operations (and No...  
Lesson on groups:    • What is a Group? | Abstract Algebra  

SOLUTION TO PRACTICE PROBLEM:

Addition on {0, 1, 2} is not a binary operation because it is not closed. Notice 2 + 2 = 4 is not in {0, 1, 2). Addition mod 3 on {0, 1, 2} is a binary operation because it is a function and it is closed. By definition the sum of any two integers mod 3 is 0, 1, or 2. Since the mod 3 sum of any two numbers in {0, 1, 2} is equal to exactly one other number in {0, 1, 2}, it is a function from {0, 1, 2}x{0, 1, 2} to {0, 1, 2} and hence is a binary operation.

Division on the rationals (Q) is not a binary operation because division is not defined on all of QxQ. Notice 0 is in Q, so elements of the form (a, 0) are in QxQ, and that is the problem. We can't divide by 0!

Multiplication on the complex numbers (C) is a binary operation. Consider two complex numbers a + bi and c + di, where i is the imaginary unit, and a, b, c, and d are real numbers. Multiplying these complex numbers and simplifying gives us (ac - bd) + (ad + bc)i. By closure of the reals under addition and subtraction, we know ac - bd and ad + bc are reals, so (ac - bd) + (ad + bc)i is a complex number. Thus, since two complex numbers multiply to exactly one other complex number, multiplication on C is a binary operation.

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Graph Theory: https://amzn.to/3JHQtZj
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Proofs and Set Theory: https://amzn.to/367VBXP (available for free online)
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Abstract Algebra: https://amzn.to/3IjoZaO
Discrete Math: https://amzn.to/3qfhoUn
Number Theory: https://amzn.to/3JqpOQd

I hope you find this video helpful, and be sure to ask any questions down in the comments!

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