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
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
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 $\left(\frac{1}{4}\cdot2\right)\in\mathcal{S}~?$