Proper subsets and arbitrary subset of the containing set

discrete mathematicselementary-set-theory

Suppose there is a set $A$ and $P$ is a proper subset of $A$. Also, suppose that $B$ is any subset (not necessarily a proper subset) of $A$. Then, are we justified in writing

$$P \subseteq B \subseteq A$$

In other words, can we be sure that it is not the case that $$B \subseteq P \subseteq A$$

Edit: $A, B, P$ are all finite sets.

Best Answer

No. For example, take $A = \{ 1, 2 \}$ and $P = \{ 1 \} \subsetneq A$.

  • If $B = \{ 1,2 \} \subseteq A$, then $P \subseteq B \subseteq A$;
  • If $B = \varnothing \subseteq A$, then $B \subseteq P \subseteq A$; and
  • If $B = \{ 2 \} \subseteq A$, then neither $B$ nor $P$ is a subset of the other, so neither $B \subseteq P \subseteq A$ nor $P \subseteq B \subseteq A$ is true.

So there isn't really much you can say about the relationship between a proper subset and a general subset.

I should also add: $P \subseteq B \subseteq A$ and $B \subseteq P \subseteq A$ are not mutually exclusive (e.g. they're both true when $P=B$), and one being false does not imply that the other is true (as the above example demonstrates), so your "in other words" is inaccurate.

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