Question

Math

Posted 5 months ago

this is amazing...

`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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Answer from Sia

Posted 5 months ago

Solution by Steps

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.

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