Let $a,b,c,d,e,f$ all be infinite sets. They are all disjoint. Assume $|a \cup b| > |c \cup d|$ and $|c \cup e| > |b \cup f|$. From this, can I conclude that $|a \cup b \cup c\cup e|>|b\cup c\cup d \cup f|$ and $|a \cup e|>|d\cup f|$?
My gut feeling says yes. Since $|a \cup b| > |c \cup d|$, there is an injective function and no bijective one from $c\cup d$ to $a\cup b$. Similarly, there is an injective function and no bijective one from $b \cup f$ to $c \cup e$. So, there must be an injective function and no bijective one from $b\cup c\cup d \cup f$ to $a \cup b \cup c\cup e$. Hence, the first inequality.
For the second inequality, I'm thinking that there must be a bijection between $b\cup c$ and itself. But since $|a \cup b \cup c\cup e|>|b\cup c\cup d \cup f|$, there must be an injective function and not bijective one from $d \cup f$ to $a \cup e$. Hence, the second inequality.
Is this right?