2.2: Binary Operation
Before beginning our formal study of groups, we need to have an understanding of binary operations. After learning to count as a child, you likely learned how to add, subtract, multiply, and divide with real numbers. As long as we avoid division by zero, these operations are examples of binary operations since we are combining two objects to obtain a single object. More formally, we have the following definition.
A binary operation \(*\) on a set \(A\) is a function from \(A\times A\) into \(A\) . For each \((a,b)\in A\times A\) , we denote the element \(*(a,b)\) via \(a*b\) . If the context is clear, we may abbreviate \(a*b\) as \(ab\) .
Don’t misunderstand the use of \(*\) in this context. We are not implying that \(*\) is the ordinary multiplication of real numbers that you are familiar with. We use \(*\) to represent a generic binary operation.
Notice that since the codomain of a binary operation on a set \(A\) is \(A\) , binary operations require that we yield an element of \(A\) when combining two elements of \(A\) . In this case, we say that \(A\) is closed under \(*\) . Binary operations have this closure property by definition. Also, since binary operations are functions, any attempt to combine two elements from \(A\) should result in a unique element of \(A\) . Moreover, since the domain of \(*\) is \(A\times A\) , it must be the case that \(*\) is defined for all pairs of elements from \(A\) .
Here are some examples of binary operations.
- The operations of \(+\) (addition), \(-\) (subtraction), and \(\cdot\) (multiplication) are binary operations on the real numbers. All three are also binary operations on the integers. However, while \(+\) and \(\cdot\) are both binary operations on the set of natural numbers, \(-\) is not a binary operation on the natural numbers since \(1-2=-1\) , which is not a natural number.
- The operation of \(\div\) (division) is not a binary operation on the set of real numbers because all elements of the form \((a,0)\) are not in the domain \(\mathbb{R}\times \mathbb{R}\) since we cannot divide by 0. Yet, \(\div\) is a suitable binary operation on \(\mathbb{R}\setminus \{0\}\) .
- Let \(A\) be a nonempty set and let \(F\) be the set of functions from \(A\) to \(A\) . Then \(\circ\) (function composition) is a binary operation on \(F\) . We utilized this fact when exploring the game Spinpossible.
- Let \(M_{2\times 2}(\mathbb{R})\) be the set of \(2\times 2\) matrices with real number entries. Then matrix multiplication is a binary operation on \(M_{2\times 2}(\mathbb{R})\) .
Let \(M(\mathbb{R})\) be the set of matrices (of any size) with real number entries. Is matrix addition a binary operation on \(M(\mathbb{R})\) ? How about matrix multiplication? What if you restrict to square matrices of a fixed size \(n\times n\) ?
Let \(A\) be a set. Determine whether \(\cup\) (union) and \(\cap\) (intersection) are binary operations on \(\mathcal{P}(A)\) (i.e., the power set of \(A\) ).
Consider the closed interval \([0,1]\) and define \(*\) on \([0,1]\) via \(a*b=\min(a,b)\) (i.e., take the minimum of \(a\) and \(b\) ). Determine whether \(*\) is a binary operation on \([0,1]\) .
Consider a square puzzle piece that fits perfectly into a square hole. Let \(R_4\) be the set of net actions consisting of the rotations of the square by an appropriate amount so that it fits back into the hole. Assume we can tell the corners of the square apart from each other so that if the square has been rotated and put back in the hole we can notice the difference. Each net action is called a symmetry of the square.
- Describe all of the distinct symmetries in \(R_4\) . How many distinct symmetries are in \(R_4\) ?
- Is composition of symmetries a binary operation on \(R_4\) ?
Let’s pause for a moment to make sure we understand our use of the word symmetry in this context. A fundamental question in mathematics is “When are two things the same?", where “things" can be whatever mathematical notion we happen to be thinking about at a particular moment. Right now we need to answer, “When do we want to consider two symmetries to be the same?" To be clear, this is a choice, and different choices can lead to different, interesting, and equally valid mathematics. For symmetries, one natural thought is that symmetries are equal when they produce the same net action on the square, meaning that when applied to a square in a particular starting position, they both yield the same ending position. In general, two symmetries are equal if they produce the same net action on the object in question.
The set \(R_4\) is called the rotation group for the square. For \(n\geq 3\) , \(R_n\) is the rotation group for the regular \(n\) -gon and consists of the rotational symmetries for a regular \(n\) -gon. As we shall see later, every \(R_n\) really is a group under composition of symmetries.
Consider a puzzle piece like the one in the previous problem, except this time, let’s assume that the piece and the hole are an equilateral triangle. Let \(D_3\) be the full set of symmetries that allow the triangle to fit back in the hole. In addition to rotations, we will also allow the triangle to be flipped over—called a reflection.
- Describe all of the distinct symmetries in \(D_3\) . How many distinct symmetries are in \(D_3\) ?
- Is composition of symmetries a binary operation on \(D_3\) ?
Repeat the above problem, but do it for a square instead of a triangle. The corresponding set is called \(D_4\) .
The sets \(D_3\) and \(D_4\) are examples of dihedral groups. In general, for \(n\geq 3\) , \(D_n\) consists of the symmetries (rotations and reflections) of a regular \(n\) -gon and is called the dihedral group of order \(2n\) . In this case, the word “order" simply means the number of symmetries in the set. Do you see why \(D_n\) consists of \(2n\) actions? As expected, we will prove that every \(D_n\) really is a group.
Consider the set \(S_3\) consisting of the net actions that permute the positions of three coins (without flipping them over) that are sitting side by side in a line. Assume that you can tell the coins apart.
- Write down all distinct net actions in \(S_3\) using verbal descriptions. Some of these will be tricky to describe. How many distinct net actions are in \(S_3\) ?
- Is composition of net actions a binary operation on \(S_3\) ?
The set \(S_3\) is an example of a symmetric group. In general, \(S_n\) is the symmetric group on \(n\) objects and consists of the net actions that rearrange the \(n\) objects. Such rearrangements are called permutations . Later we will prove that each \(S_n\) is a group under composition of permutations.
Explain why composition of spins is not a binary operation on the set of spins in \(\text{Spin}_{3\times 3}\) .
Some binary operations have additional properties.
Let \(A\) be a nonempty set and let \(*\) be a binary operation on \(A\) .
- We say that \(*\) is associative if and only if \((a*b)*c=a*(b*c)\) for all \(a,b,c\in A\) .
- We say that \(*\) is commutative if and only if \(a*b=b*a\) for all \(a,b\in A\) .
Provide an example of each of the following.
- A binary operation on a set that is commutative.
- A binary operation on a set that is not commutative.
Provide an example of a set \(A\) and a binary operation \(*\) on \(A\) such that \((a*b)^2\neq a^2*b^2\) for some \(a,b\in A\) . Under what conditions will \((a*b)^2= a^2*b^2\) for all \(a,b\in A\) ? Note: The notation \(x^2\) is shorthand for \(x*x\) .
Define the binary operation \(*\) on \(\mathbb{R}\) via \(a*b=1+ab\) . In this case, \(ab\) denotes the multiplication of the real numbers \(a\) and \(b\) . Determine whether \(*\) is associative on \(\mathbb{R}\) .
[thm:function_comp_associative] If \(A\) is a nonempty set and \(F\) is the set of functions from \(A\) to \(A\) , then function composition is an associative binary operation on \(F\) .
When the set \(A\) is finite, we can represent a binary operation on \(A\) using a table in which the elements of the set are listed across the top and down the left side (in the same order). The entry in the \(i\) th row and \(j\) th column of the table represents the output of combining the element that labels the \(i\) th row with the element that labels the \(j\) th column (order matters).
Consider the following table.
\(\begin{array}{c|c|c|c}
* & a & b & c \\
\hline a & b & c & b \\
\hline b & a & c & b \\
\hline c & c & b & a
\end{array}\)
This table represents a binary operation on the set \(A=\{a,b,c\}\). In this case, \(a*b=c\) while \(b*a=a\). This shows that \(*\) is not commutative.
Consider the following table that displays the binary operation \(*\) on the set \(\{x,y,z\}\) .
\(\begin{array}{c|c|c|c}
* & x & y & z \\
\hline x & x & y & z \\
\hline y & y & x & x \\
\hline z & y & x & x
\end{array}\)
- Determine whether \(*\) is commutative.
- Determine whether \(*\) is associative.
What property must the table for a binary operation have in order for the operation to be commutative?