SECTION 2 BINARY OPERATIONS Definition: A binary operation  on a set S is a function mapping S X S into S. For each (a, b)  S X S, we will denote the element  ((a, b)) of S by a  b. Example: Our usual addition + is a binary operation on the set R (C, Z, R +, or Z + ). Our usual multiplication  is a binary operation on R (C, Z, R +, or Z + ). On Z +, we define a binary operation  by a  b equals the smaller of a and b, or the common value of a  b. On Z +, we define a binary operation  ’ by a  ’ b = a.

Induced Operation Definition Let  be a binary operation on S and let H be a subset of S. The subset H is closed under  if for all a, b  H, we also have a  b  H. In this case, the binary operation on H given by restricting  to H is the induced operation of  on H. Example: Our usual addition + on the set R of real numbers does not induce a binary operation on the set R * of nonzero real numbers. Since 2  R * and - 2  R *, but 2+(-2)=0 and 0  R *. Thus R * is not closed under .

In order to define a binary operation  on a set S we must be sure that the following two conditions hold:  is everywhere defined on S That is, for each possible ordered pair of elements in S, there is exactly one element assigned to it. S is closed under  That is, for each ordered pair of elements in S, the element assigned to it is again in S.

Examples On Q, let a  b = a / b. Such  is not everywhere defined on Q. Since no rational number is assigned by this rule to the pair (2, 0). On Q +, let a  b = a / b. Such  is a binary operation on Q + since it satisfies two previous conditions. On Z +, let a  b = a / b. Such  is not a binary operation on Z +. Since Z + is not closed under . Ex: 1  3 is not in Z +. Exercise: 17-19

Commutative & Associative Definition A binary operation  on a set S is commutative if a  b = b  a Definition A binary operation on a set S is associative if (a  b)  c = a  (b  c) For all a, b, c  S.

Tables For a finite set, a binary operation on the set can be defined by means of a table in which the elements of the set are listed across the top as heads of columns and at the left side as heads of rows. We always require that the elements of the set be listed as heads across the top in the same order as heads down the left side.

Example The following table defines the binary operation  on S ={a, b, c} by the following rule: (ith entry on the left)  (jth entry on the top) = (entry in the ith row and jth column of the table body).  a b c a b c b b a c b c c b a Thus a  b=c and b  a=a, so  is not commutative.

Note: a binary operation defined by a table is commutative if and only if the entries in the table are symmetric with respect to the diagonal that starts at the upper left corner of the table and terminates at the lower right corner. For example: Complete the following table so that  is a commutative binary operation on the set S = {a, b, c, d}.  a b c d a b b d a c a c d d a b b c Exercise 1- 5