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Thread: Groups and sets questions

  1. #1
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    Post Groups and sets questions

    S is subset of R S.T x+y∈S for all x,y∈S

    Is ax ∈ S for all s ∈ S, a ∈R? Prove it
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  2. #2
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    Re: Groups and sets questions

    Supposing R denotes the set of real numbers.
    Supposing "ax ∈ S for all s ∈ S, a ∈R?" is a typo and should be "a*s ∈ S for all s ∈ S, a ∈R?"
    Supposing it's for all a ∈ R since question didn't specify there exists a ∈ R
    If any of the above suppositions doesn't hold, proof below is invalid.

    The statement that you are being told to prove is false (again, if the statement is for all a ∈ R.)
    This can be proved by counterexample. Consider S = {0, 1, 2, 3, ... } and a = 1.5
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  3. #3
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    Re: Groups and sets questions

    Quote Originally Posted by Mustafashama View Post
    S is subset of R S.T x+y∈S for all x,y∈S
    Is ax ∈ S for all s ∈ S, a ∈R? Prove it
    Presumably $\mathbb{R}$ is the group of real numbers with the additive operation.
    If $\mathcal{S}=\mathbb{Z}\subset\mathbb{R}$ then $(\forall m~\&~n\in\mathcal{S})[m+n\in\mathcal{S}]$
    Now $\frac{1}{4}\in\mathbb{R}~\&~2\in\mathcal{S}$ is it the case that $\left(\frac{1}{4}\cdot2\right)\in\mathcal{S}~?$
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