Let $X\neq \emptyset$ be a set, let $\text{Part}(X)$ be the set of partitions of $X$ (where we require that $\emptyset \notin P$ whenever $P\in\text{Part}(X)$).

For $P, Q\in \text{Part}(X)$ we say that $P$ covers $X$ more efficiently than $Q$ if $$\text{card}(P\setminus Q) < \text{card}(Q\setminus P), $$ and we write $P<_{\text{eff}}Q$ for this.

Is $<_{\text{eff}}$ transitive?