1/12/2024 0 Comments Empty family feud set![]() ![]() It leaves $\cap \emptyset$ as an untidy loose end, which they may later trip over. This option works perfectly well, but some logicians dislike it. How does one define $a/0$ ? One option is to leave $\cap \emptyset$ undefined, since there is no very satisfactory way of defining it. This presents a mild notational problem: How do we define $\cap \emptyset$ ? The situation is analogous to division by zero in arithmetic. Since the right side is true of every $x$. $x \in C \Leftrightarrow x$ belongs to every member of $\emptyset$ By Theorem 2A, there is no set $C$ such that for all $x$, To which $x$ fails to belong.) Thus it looks as if $\cap \emptyset$ should be the class $V$ What happens if $A = \emptyset$ ? For any $x$ at all, it is vacuously true that $x$ belongs to every member of $\emptyset$. $x \in \cap A \Leftrightarrow x$ belongs to every member of $A$. In general, we define for every non empty set $A$, the intersection $\cap A$ of $A$ Suppose we want to take the intersection of infinitely many sets $b_0, b_1, \ldots$. ![]() See Herbert Enderton, Elements of Set Theory (1977), page 24 : ![]()
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