Subset Condititonal Question

[Zarlucicil]

I've used the following implication (conditional...whatever you want to call it) in a few proofs and was wondering if it's actually is true. I incorporated it into my proofs because it seemed to make obvious sense, but I'm not sure if I'm overlooking something- obvious or subtle.

$$T \subseteq S \Rightarrow \exists s' \in S \& \exists s'' \in S \ni [s' \leq t \leq s''], \forall t \in T$$.

English: If T is a subset of S, then there exists an s' in S and an s'' in S such that t is greater than or equal to s' and less than or equal to s'', for all t in T.

 

[HallsofIvy]

First, we are not talking about general sets. In order for the inequalities to make sense, S must be a linearly ordered set- probably the set or real numbers. And it looks to me like, in order for that statement to be true, S and T must be intervals specifically.

 

[Zarlucicil]

S must be a linearly ordered set

Yes- I'm sorry. S is a subset of the real numbers.

And it looks to me like, in order for that statement to be true, S and T must be intervals specifically.

I suppose that might be true, but I can't think of a counterexample involving non-interval sets nor have I found a way to disprove the implication for non-interval sets. It seems to be true for at least some non-interval sets. For example, when T = {-3.2, -1, 7} and S = {-4, -3.2, -1, 0, 7, 9}. Hmm, or are these example sets considered to be "intervals" because they can be written as the union of intervals? --> T = [-3.2, -3.2] U [-1, -1] U [7, 7]. If they are considered to be intervals, then I don't know what wouldn't be considered an interval.

 

[zut837]

This is true for any ordered set S. Just pick s'=s''=t

 

[blue_raver22]

I can confirm that this implication is indeed true. It is a fundamental property of subsets that if T is a subset of S, then all elements in T must also be elements of S. This means that there must exist some elements in S that are both greater than or equal to the smallest element in T (s') and less than or equal to the largest element in T (s''). This is precisely what the implication states.

It is also worth noting that this implication is not limited to proofs, but can also be applied in practical situations. For example, in a set of data, if one subset is a subset of another, then all data points in the smaller subset must also be present in the larger subset. This can be useful in data analysis and comparisons.

In summary, the subset conditional question you have used in your proofs is a valid implication and can be applied in various scenarios, including practical ones.