I've been reading through Axiomatic Set Theory by Patrick Suppes and stumbled upon this symbol, ≺, on page 97. The definition reads as follows: We now define in the expected manner the relation ≺ of having less power. It then shows an example theorem stating to prove A ≺ A ∪ B, but I don't understand what the author means by having less power.
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5 - 2$\begingroup$ Since have the book, and many people do not, you could help us help you by giving the exact definition. $\endgroup$Thomas Andrews– Thomas Andrews2016-01-22 01:07:24 +00:00Commented Jan 22, 2016 at 1:07
- $\begingroup$ My guess would be $A < B$ is equivalent to "$A$ is a subset of $B$." $\endgroup$AJY– AJY2016-01-22 01:15:30 +00:00Commented Jan 22, 2016 at 1:15
- 4$\begingroup$ "Power" is sometimes used as a synonym for "cardinality". $\endgroup$Eric Wofsey– Eric Wofsey2016-01-22 01:16:06 +00:00Commented Jan 22, 2016 at 1:16
- 1$\begingroup$ All that is clear when taken out of context is that $\prec$ is generally used as a partial order or a strict partial order. See wikipedia. In the case that they intend $\prec$ to be the strict partial order defined as $A\prec B\Leftrightarrow |A|<|B|$, then the statement $A\prec A\cup B$ is not true when $B\subseteq A$ or when $A$ is infinite in size. It will however be true if it is not a strict partial order, i.e. $A\prec B\Leftrightarrow |A|\leq |B|$ $\endgroup$JMoravitz– JMoravitz2016-01-22 01:32:15 +00:00Commented Jan 22, 2016 at 1:32
- 3$\begingroup$ This is a legitimate question and does not deserve to be closed. Specially not after less than an hour! $\endgroup$lhf– lhf2016-01-22 01:51:58 +00:00Commented Jan 22, 2016 at 1:51
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1 Answer
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2 On page 94 it says:
I guess $A\approx C$ means that $A$ and $C$ have the same cardinality, and so $A \preceq B$ means that there is an injection $A \to B$.
Then on page 97 it says:
It is clear that $A \prec B$ does not mean $A \subset B$. Using the interpretation I gave above for $A \preceq B$, it probably means that the cardinality of $A$ is strictly less than the cardinality of $B$, in the sense that there is an injection $A \to B$ but not an injection $B \to A$.
"$A$ is a subset of $B$" is defined on page 22 and uses the standard notation $A \subseteq B$.
