Section 2 – Binary Operations Instructor: Yifan Yang Fall 2006 Outline Binary operation Definition and examples Commutativity and associativity Tables for binary operations Binary operations Definition (2.1) A binary operation ∗ on a set S is a function mapping S × S into S. For each (a, b) ∈ S × S, we denote the element ∗((a, b)) of S by a ∗ b. Example 1. The usual addition + is a binary operation on the set R, + and also on the sets Z, Q, Z , and C. 2. The usual division / is not a binary operation on R since / is not defined for the pair (a, 0) ∈ R × R. It is not a binary + + operation on Z either since a/b may not be in Z . 3. The usual matrix addition + is not a binary operation on the set M(R) of real matrices since A + B is defined only when the sizes are the same. Examples of binary operations Example 1. The function ∗ : Z × Z 7→ Z defined by a ∗ b = min(a, b) is a binary operation on Z. 2. Let F be the set of all real-valued functions f : R 7→ R. Then +F , −F , ·F , and ◦ on F × F defined by (f +F g)(x) = f (x) + g(x) (f −F g)(x) = f (x) − g(x) (f ·F g)(x) = f (x) · g(x) (f ◦ g)(x) = f (g(x)) are all binary operations on F. Induced operation Definition (2.4) Let ∗ be a binary operation on a set S and let H be a subset of S. If for all a, b ∈ H we also have a ∗ b ∈ H, then H is closed under ∗. In this case, the binary operation on H given by restricting ∗ to H is the induced operation of ∗ on H. Example 1. We have Z ⊂ Q ⊂ R ⊂ C. The addition + on C induces a binary operation on the above sets. 2. The usual addition + on C does not induce a binary ∗ ∗ operation on C since C is not closed under addition. (We ∗ ∗ have 2, −2 ∈ C , but 0 = 2 + (−2) 6∈ C .) Induced operation Example Consider the set Z and the subset H = {n2 : n ∈ Z}. Then H is not closed under addition since 1, 4 ∈ H, but 1 + 4 6∈ H. But H is closed under multiplication because if r, s ∈ H, then r = n2, s = m2 for some integers m and n, and rs = (mn)2 ∈ H. Commutativity of binary operations Definition A binary operation ∗ on a set S is commutative if a ∗ b = b ∗ a for all a, b ∈ S. Example 1. The usual +, · on Z, Q, R, and C are all commutative. 2. The usual − on Z is not commutative. 3. The composition operation ◦ on the set of all real-values functions f : R 7→ R is not commutative. (For example, if f (x) = x2, g(x) = x + 1, then f ◦ g(x) = (x + 1)2, but g ◦ f (x) = x2 + 1. 4. The matrix multiplication on the set of all n × n matrices is not commutative. Associativity of binary operations Definition A binary operation ∗ on a set S is associative if (a ∗ b) ∗ c = a ∗ (b ∗ c) for all a, b, c ∈ S. Example 1. The usual + and · on C are both associative. 2. The usual − on C is not associative. 3. The matrix multiplication is associative. 4. The composition operation ◦ on the set of all real-valued functions f : R 7→ R is associative. Tables for binary operations When a set S has a finite number of elements, we can describe binary operations on S by means of tables. For example, for Z5, the tables for +5 and ·5 are ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ +5 0 1 2 3 4 ·5 0 1 2 3 4 0¯ 0¯ 1¯ 2¯ 3¯ 4¯ 0¯ 0¯ 0¯ 0¯ 0¯ 0¯ 1¯ 1¯ 2¯ 3¯ 4¯ 0¯ 1¯ 0¯ 1¯ 2¯ 3¯ 4¯ 2¯ 2¯ 3¯ 4¯ 0¯ 1¯ 2¯ 0¯ 2¯ 4¯ 1¯ 3¯ 3¯ 3¯ 4¯ 0¯ 1¯ 2¯ 3¯ 0¯ 3¯ 1¯ 4¯ 2¯ 4¯ 4¯ 0¯ 1¯ 2¯ 3¯ 4¯ 0¯ 4¯ 3¯ 2¯ 1¯ Tables for binary operations Observation The above two tables are symmetric with respect to the diagonal. This is because +5 and ·5 are commutative. In general, if ∗ is commutative, then the table is symmetric with respect to the diagonal. In-class exercises Is the definition of the italicized term correct? If not, correct it. 1. A binary operation ∗ is commutative if and only if a ∗ b = b ∗ a. 2. A binary operation ∗ on a set S is associative if and only if, for all a, b, c ∈ S, we have (b ∗ c) ∗ a = b ∗ (c ∗ a). 3. A subset H of a set S is closed under a binary operation ∗ on S if and only if (a ∗ b) ∈ H for all a, b ∈ S. In-class exercise Determine whether the following binary operations are commutative and whether they are associative. + 1. ∗ on Z by a ∗ b = 2ab. 2. ∗ on Q by a ∗ b = ab/2. 3. Let S = {a, b, c, d}. Let the binary operation ∗ on S be given as ∗ a b c d a b d a a b d a c b c a c d b d a b b c Homework Homework Do Problems 8, 12, 20, 22, 24, 28, 36, 37 of Section 2..