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Question on cardinality of sets

Mathematics Asked by jshthng on December 6, 2021

Let $a=bigcup_{iin I_a}a_i$ and $b=bigcup_{jin I_b}b_j$ where $I_a$ is the index set of $a$ and $I_b$ is the index set of $b$, such that

  1. $forall iin I_a:a_i$ is a finite set
  2. $forall jin I_b:b_j$ is a finite set
  3. $a_kcap a_m= b_lcap b_n=emptyset$ if $kneq m$ and $lneq n$
  4. $left |a_i right |geleft |b_j right |$ for every $iin I_a$ and $jin I_b$
  5. $left |I_a right |>left |I_b right |$.

Is $left |a right |>left |b right|$?

One Answer

Condition 3 is phrased in a very strange way and should ring alarm bells. If we let $$I_a={i_1,i_2}quad {rm and} quad I_b={j_1}$$ then we can have $$ a_{i_1}={1,2,3},qquad a_{i_2}={1,2,3},qquad b_{j_1}={4,5,6}.$$

This satisfies all of the given conditions. Do you see why it satisfies condition 3 despite appearances? None the less $|a|=|b|$.

Answered by tkf on December 6, 2021

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