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Definition. A binary operation * on a non-empty set S is a rule that assigns to each ordered pair... more Definition. A binary operation * on a non-empty set S is a rule that assigns to each ordered pair of elements of elements of S a uniquely determined element of S. The element assigned to the ordered pair (a, b) with a, b ∈ S is denoted by a * b. Remark. In other words, a binary operarion of a set S is a function * : S×S → S from the Cartesian product S × S to the set S. The only difference is that the value of the function * at an ordered pair (a, b) is denoted by a * b rather than * ((a, b)). Examples. Let S = N = {1, 2, 3,. . .}. (1) a ⋆ b = max(a, b) e.g. 2 ⋆ 3 = 3, 3 ⋆ 2 = 3, 3 ⋆ 3 = 3. (2) a ⋄ b = a e.g. 2 ⋄ 3 = 2, 3 ⋄ 2 = 3, 3 ⋄ 3 = 3. (3) ab = a b e.g. 23 = 2 3 = 8, 32 = 3 2 = 9, 33 = 3 3 = 27.
Definition. A binary operation * on a non-empty set S is a rule that assigns to each ordered pair... more Definition. A binary operation * on a non-empty set S is a rule that assigns to each ordered pair of elements of elements of S a uniquely determined element of S. The element assigned to the ordered pair (a, b) with a, b ∈ S is denoted by a * b. Remark. In other words, a binary operarion of a set S is a function * : S×S → S from the Cartesian product S × S to the set S. The only difference is that the value of the function * at an ordered pair (a, b) is denoted by a * b rather than * ((a, b)). Examples. Let S = N = {1, 2, 3,. . .}. (1) a ⋆ b = max(a, b) e.g. 2 ⋆ 3 = 3, 3 ⋆ 2 = 3, 3 ⋆ 3 = 3. (2) a ⋄ b = a e.g. 2 ⋄ 3 = 2, 3 ⋄ 2 = 3, 3 ⋄ 3 = 3. (3) ab = a b e.g. 23 = 2 3 = 8, 32 = 3 2 = 9, 33 = 3 3 = 27.