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GO Classes Test Series 2024 | Discrete Mathematics | Test 3 | Question: 20

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Let $\text{R}$ be the set of all real numbers.

Let $\text{S} = \text{R}\setminus \{-1\}$ and define a binary operation on $S$ by $a \ast b = a + b + ab.$

Which of the following is true?

  1. $(\text{R},\ast)$ is an abelian group, but $(\text{S},\ast)$ is not.
  2. $(\text{S},\ast)$ is an abelian group, but $(\text{R},\ast)$ is not.
  3. Both $(\text{R},\ast),(\text{S},\ast)$ are abelian groups.
  4. None of $(\text{R},\ast),(\text{S},\ast)$ are abelian groups.
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Both $(\text{R},\ast)$ and $(\text{S},\ast)$ are monoids, but $(\text{R},\ast)$ is not a group as for “$-1$” there is no inverse.


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GO Classes Test Series 2024 | Discrete Mathematics | Test 3 | Question: 21
In a group $\text{G},$ every element other than the identity element has order $2$ then $\text{G}$ is? Abelian group Cyclic group Non-abelian group Non-cyclic group
In a group $\text{G},$ every element other than the identity element has order $2$ then $\text{G}$ is?Abelian groupCyclic groupNon-abelian groupNon-cyclic group
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