The word “binary” means consisting of two pieces and a binary operation is a rule for merging two values to obtain a new value. The operations such as addition and multiplication that we all learned in elementary school are actually the most common binary operations.

 

In order to understand binary operations, we first need to look at the definition of a mapping which is very similar to a function.

 

A relationship between the sets X and Y where each              is the first component of exactly one ordered pair (x, y) in      is defined as the function     mapping X into Y. A mapping of X into Y is the name given for these kinds of functions. We write                                and show                      by                   . The set X is the domain; the set Y is the codomain, and                                       

is the range of       . 

 

The addition of real numbers can be viewed as the function                           , which is the mapping of             into      . For instance, the action of + on                          can be showed in function notation by +((3,4)) = 7. To show this in set notation, we write                           . In reality, all these expressions are the same as our familiar notation 3 + 4 = 7.

 

Now, we are ready to investigate binary operations.

 

A binary operation      on a set S is                             . For each                           , we will indicate the element                 of S by           .

 

Addition and multiplication are binary operations on the set ℝ.

 

Let       be a binary operation on a set S, and let H be a subset of S.  If we have                   for all                , we say that the subset H is closed under    . For this case, the induced operation of     on H is the binary operation on H that is provided by limiting        to H. 

 

The following example will help the reader to understand this concept better.

 

On the set ℤ, let + and     be the binary operations of addition and multiplication, and let                                . We will now check to see if H is closed under multiplication and addition.

 

For addition, we only need to observe that             and             are in H, but that 4 + 9 = 13 and            . Hence, we can conclude that H is not closed under addition.

 

For multiplication, let              and           . In order for H to include r and s, there should be integers n and m in        such that                  and             . We can easily observe that                                    . Based on the definition of elements in H and the fact that                , we can conclude that                which means H is closed under multiplication.

 

A binary operation on a set S is associative if                                         and commutative if                        for all                 .

 

Let’s define a binary operation     on       where          is equal to the larger of a and b and equal to 1 if a = b. From this definition                                        and                                      , which shows it is not hard to see that this binary operation     is associative. Similarly, if we observe that                     and                  we can easily come to the conclusion that this binary operation       is commutative.

 

References:

Fraleigh, John B. “A First Course in Abstract Algebra”. Essex, Pearson Education Limited, 2014.

Roberts, D. R. and F. (n.d.). Binary Operations - MathBitsNotebook(A1 - CCSS Math). https://mathbitsnotebook.com/Algebra1/RealNumbers/RNBinary.html. 

Wikipedia contributors. "Function (mathematics)." Wikipedia, The Free Encyclopedia. Wikipedia, The Free Encyclopedia, 14 May. 2021. Web. 17 May. 2021.

 

Figure References:

[1] Wikipedia contributors. "Binary operation." Wikipedia, The Free Encyclopedia. Wikipedia, The Free Encyclopedia, 6 Jan. 2021. Web. 17 May. 2021.