set theory - cardinality of product modulo direct sum

The cardinality of the reduced product is always the same as that of the product, modulo omitting finitely many unusually large $X_i$'s. (Even when $I$ is finite, in which case all $X_i$'s should be omitted.)

Arrange the sets $X_i$ in nondecreasing order of size in a wellordered sequence $(X_alpha)_{alpha<tau}$. We may assume that $tau$ is a limit ordinal. Otherwise, the last coordinate $tau-1$ (like any single coordinate) contributes nothing to the reduced product. Omit $X_{tau-1}$ and repeat as long as necessary.

I will show that then $|X| = |X/{sim}|$ where $X = prod_{alpha<tau} X_alpha$.

Let $kappa = sup_{alpha<tau} |X_alpha|$. Since each ${sim}$-equivalence class has size $|tau|cdotkappa$ (see note) we have

$|X| leq |X/{sim}|cdot|tau|cdotkappa = max(|X/{sim}|,|tau|,kappa).$

Since $|X| geq 2^{|tau|} > |tau|$, we conclude that either $|X/{sim}| = |X|$ or $|X/{sim}| leq |X| leq kappa$.

Since $tau$ is a limit ordinal, the diagonal embedding $d:kappa to prod_{alpha<tau} |X_alpha|$, where

$d_alpha(xi) = begin{cases} xi & when xi < |X_alpha| \ 0 & otherwise,end{cases}$

shows that $kappa leq |X/{sim}|$. So, in the case $|X/{sim}| leq |X| leq kappa$, we in fact have $|X/{sim}| = |X| = kappa$.

Note: The elements ${sim}$-equivalent to a given $x in X$ are obtained by selecting finitely many new values from the sets $X_alpha-{x_alpha}$ to replace the corresponding value of the sequence $x$. There are at least $sum_{alpha<tau} |X_alpha-{x_alpha}|$ and no more than $left(sum_{alpha<tau} |X_alpha|right)^{<omega}$ ways of doing this. Since $tau$ is infinite and the $X_alpha$'s all have two or more elements, these two bounds are equal to $sum_{alpha<tau} |X_alpha| = |tau|cdotkappa$.