Define a binary operation on Z, the set of integers, by the equation m • n = m + n + mn. Which of the following statements is (are) true about the binary structure (Z, • )? I. This structure is not a group since the operation is not associative. II. This structure is not a group since there is no identity element. III. This structure is not a group since not all elements have an inverse. IV. This structure is a group.[ "I and II only", "I and III only", "II and III only", "III only" ]

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∻Solution by Steps∻ ‖ step 1 ⋮ To determine the properties of the binary operation defined by $ m \cdot n = m + n + mn $, we first check for associativity. ‖ ‖ step 2 ⋮ We need to check if $ (m \cdot n) \cdot p = m \cdot (n \cdot p) $ for all integers $ m, n, p $. ‖ ‖ step 3 ⋮ Calculating $ (m \cdot n) \cdot p $ gives us $ (m + n + mn) \cdot p = m + n + mn + p + (m + n + mn)p $. ‖ ‖ step 4 ⋮ Calculating $ m \cdot (n \cdot p) $ gives us $ m \cdot (n + p + np) = m + (n + p + np) + m(n + p + np) $. ‖ ‖ step 5 ⋮ Comparing both expressions from step 3 and step 4, we see that they are not equal in general, so the operation is not associative. ‖ ‖ step 6 ⋮ To find an identity element $ e $, we need an element such that $ m \cdot e = m $ for all integers $ m $. ‖ ‖ step 7 ⋮ Solving $ m + e + me = m $ leads to $ e + me = 0 $. The only integer that satisfies this for all $ m $ is $ e = 0 $. However, substituting $ e = 0 $ into the equation gives $ m \cdot 0 = m + 0 + m \cdot 0 = m + m \cdot 0 $, which does not simplify to $ m $ unless $ m = 0 $. Thus, there is no identity element. ‖ ‖ step 8 ⋮ To find an inverse for an element $ m $, we need an element $ n $ such that $ m \cdot n = e $, where $ e $ is the identity element. Since there is no identity element, not all elements can have an inverse. ‖ ***C*** ∻Key Concept∻ ⚹Group Properties⚹ ∻Explanation∻ ⚹A group requires associativity, an identity element, and inverses for all elements. The binary structure (Z, • ) lacks associativity and an identity element, and therefore not all elements have inverses.⚹◊Is the binary structure (Z, • ) a group under the given operation?⍭ Generate me a similar question◊

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