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"Define $ϕ:A→B×C,\ g(H∩K)↦(gH,gK)$.

Note that $ϕ$ is well defined and also it's trivial that it is injective.

Thus, $|A|≤|B||C|$."

is from the question here.

But my question is, why does the inequality follow from the fact that the map is injective? I'm very confused.

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Nate
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1 Answers1

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One of the definitions of cardinality is that $ |A| \leq |B| $ is equivalent to the existence of an injective mapping $ A \to B$. So inequality follows directly from the definition of cardinality.

user26857
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Q the Platypus
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