Why isn't $< \{ 1\} \{2 \}>$ contained in $< \{1,2\} \{3,4\}>$?

Question

Why isn't subsequence $< \{ 1\} \{2 \}>$ contained in sequence $< \{1,2\} \{3,4\}>$?

Since: $\{1\} \subseteq \{1,2\}$ and $\{2\} \subseteq \{1,2\}$?

Is it because they belong to the same set (that's, they index into the same set) that $<\{ 1\} \{2 \}>$ isn't a subsequence of the given sequence?

Comparatively,

$$<\{2\} \{4\}>$$ is contained in

$$<\{2,4\}\{2,4\}\{2,5\}>$$

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A sequence $<a_1...a_n>$ is contained in $<b_1,...,b_m>$ if exists integers $i_1 < ... < i_n$ such that $a_1 \subseteq b_{i1}, ..., a_n \subseteq b_{in}$.

Answer 1

A sequence $<a_1...a_n>$ is contained in $<b_1,...,b_m>$ if exists integers $i_1<...<i_n$ such that $a_1⊆b_{i_1},...,a_{n}⊆b_{i_n}$.

That answers your question doesn't it?

$b_1 = \{1\}\subset a_1$ but $b_2 = \{2\} \subset a_1$. And $1 = 1$. It is not the case that $1 < 1$.

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So by your definition, $<\{1\},\{3\}>$ is contained.

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I must say I have NEVER heard of your definition of a sequence of subsets being "contained" in a sequence of sets. And I must say I don't like it. It is way too ambiguous and sounds too much like being a subsequence-- which it is nothing of the sort.