Editing Binary operation (section)

== Properties and examples ==
Typical examples of binary operations are the [[addition]] (<math>+</math>) and [[multiplication]] (<math>\times</math>) of [[number]]s and [[matrix (mathematics)|matrices]] as well as [[composition of functions]] on a single set.
For instance,
* On the set of real numbers <math>\mathbb R</math>, <math>f(a,b)=a+b</math> is a binary operation since the sum of two real numbers is a real number.
* On the set of natural numbers <math>\mathbb N</math>, <math>f(a,b)=a+b</math> is a binary operation since the sum of two natural numbers is a natural number.  This is a different binary operation than the previous one since the sets are different.
* On the set <math>M(2,\mathbb R)</math> of <math>2 \times 2</math> matrices with real entries, <math>f(A,B)=A+B</math> is a binary operation since the sum of two such matrices is a <math>2 \times 2</math> matrix.
* On the set <math>M(2,\mathbb R)</math> of <math>2 \times 2</math> matrices with real entries, <math>f(A,B)=AB</math> is a binary operation since the product of two such matrices is a <math>2 \times 2</math> matrix.
* For a given set <math>C</math>, let <math>S</math> be the set of all functions <math>h \colon C \rightarrow C</math>.  Define <math>f \colon S \times S \rightarrow S</math> by <math>f(h_1,h_2)(c)=(h_1 \circ h_2)(c)=h_1(h_2(c))</math> for all <math>c \in C</math>, the composition of the two functions <math>h_1</math> and <math>h_2</math> in <math>S</math>. Then <math>f</math> is a binary operation since the composition of the two functions is again a function on the set <math>C</math> (that is, a member of <math>S</math>).

Many binary operations of interest in both algebra and formal logic are [[commutative]], satisfying <math>f(a,b)=f(b,a)</math> for all elements <math>a</math> and <math>b</math> in <math>S</math>, or [[associative]], satisfying <math>f(f(a,b),c)=f(a,f(b,c))</math> for all <math>a</math>, <math>b</math>, and <math>c</math> in <math>S</math>.  Many also have [[identity element]]s and [[inverse element]]s.

The first three examples above are commutative and all of the above examples are associative.

On the set of real numbers <math>\mathbb R</math>, [[subtraction]], that is, <math>f(a,b)=a-b</math>, is a binary operation which is not commutative since, in general, <math>a-b \neq b-a</math>.  It is also not associative, since, in general, <math>a-(b-c) \neq (a-b)-c</math>; for instance, <math>1-(2-3)=2</math> but <math>(1-2)-3=-4</math>.

On the set of natural numbers <math>\mathbb N</math>, the binary operation [[exponentiation]], <math>f(a,b)=a^b</math>, is not commutative since, <math>a^b \neq b^a</math> (cf. [[Equation x^y = y^x|Equation x<sup>y</sup> = y<sup>x</sup>]]), and is also not associative since <math>f(f(a,b),c) \neq f(a,f(b,c))</math>.  For instance, with <math>a=2</math>, <math>b=3</math>, and <math>c=2</math>, <math>f(2^3,2)=f(8,2)=8^2=64</math>, but <math>f(2,3^2)=f(2,9)=2^9=512</math>.  By changing the set <math>\mathbb N</math> to the set of integers <math>\mathbb Z</math>, this binary operation becomes a partial binary operation since it is now undefined when <math>a=0</math> and <math>b</math> is any negative integer.  For either set, this operation has a ''right identity'' (which is <math>1</math>) since <math>f(a,1)=a</math> for all <math>a</math> in the set, which is not an ''identity'' (two sided identity) since <math>f(1,b) \neq b</math> in general.

[[division (mathematics)|Division]] (<math>\div</math>), a partial binary operation on the set of real or rational numbers, is not commutative or associative.  [[Tetration]] (<math>\uparrow\uparrow</math>), as a binary operation on the natural numbers, is not commutative or associative and has no identity element.