Set difference

From CS2800 wiki

Definition: Set difference
If [math]A [/math] and [math]B [/math] are sets, then the set difference [math]A [/math] minus [math]B [/math] (written [math]A \href{/cs2800/wiki/index.php/%5Csetminus}{\setminus} B [/math]) is given by [math]A \href{/cs2800/wiki/index.php/%5Csetminus}{\setminus} B \href{/cs2800/wiki/index.php/Definition}{:=} \{x \href{/cs2800/wiki/index.php/%5Cmid}{\mid} x \href{/cs2800/wiki/index.php/%5Cin}{\in} A \href{/cs2800/wiki/index.php/And}{\text{ and }} x \href{/cs2800/wiki/index.php/%5Cnotin}{\notin} B\} [/math].

We use the symbol [math]\href{/cs2800/wiki/index.php/%5Csetminus}{\setminus} [/math] instead of the normal [math]- [/math] because occasionally we will want to use sets to represent number-like things, and we will want to define subtraction differently for those sets (in particular, we will do this in the section on Category:number theory).

This means that if you know [math]x \href{/cs2800/wiki/index.php/%5Cin}{\in} A \href{/cs2800/wiki/index.php/%5Csetminus}{\setminus} B [/math], you can conclude 'both' that [math]x \href{/cs2800/wiki/index.php/%5Cin}{\in} A [/math] 'and' [math]x \href{/cs2800/wiki/index.php/%5Cnotin}{\notin} B [/math], and similarly you must prove both [math]x \href{/cs2800/wiki/index.php/%5Cin}{\in} A [/math] and [math]x \href{/cs2800/wiki/index.php/%5Cnotin}{\notin} B [/math] to prove [math]x \href{/cs2800/wiki/index.php/%5Cin}{\in} A \href{/cs2800/wiki/index.php/%5Csetminus}{\setminus} B [/math].

In this Venn diagram, the set difference [math]A [/math] minus [math]B [/math] is shaded: