《Discrete Mathematics II教学-华南理工》Lecture 2. Binary Operations.pdfVIP

I.2 Binary Operations 1 Section I.2. Binary Operations Note. In this section, we deal abstractly with operations on pairs (thus the term “binary”) of elements of a set. You are familiar with this concept in the settings of addition, subtraction, multiplication, and (except for 0) division of numbers. Two numbers, such as 9 and 3, yield through these four operations, the numbers 12, 6, 27, and 3, respectively. Notice that taking the 9 first and the 3 second affects the result for subtraction and division. That is, order matters for these operations. Definition. A binary operation ∗ on a set S is a function mapping S × S into S . For each (ordered pair) (a, b) ∈ S ×S , we denote the element ∗((a, b)) ∈ S as a ∗ b. Example. The easiest examples of binary operations are addition and multiplica- tion on R. We could also consider these operations on different sets, such as Z, Q, or C. Note. As we’ll see, we don’t normally think of subtraction and division as bi- nary operations, but instead we think of them in terms of manipulation of inverse elements with respect to addition and multiplication (respectively). Example. A more exotic example of a binary operation is matrix multiplication on the set of all 2 × 2 matrices. Notice that “order matters” (and there is, in general, no such thing as “division” here). I.2 Binary Operations 2 Definition 2.4. Let ∗ be a binary operation on set S and let H ⊆ S . Then 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. Let E = {n ∈ Z | n is even} and let O = {n ∈ Z | n is odd}. Then, E is closed under addition (and multiplication). However