Question on cardinality of sets

Question

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

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

tkf · 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 $|a|=|b|$.

Question on cardinality of sets

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