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→Application -- Combinatorics: no need for formal logic symbols in a basic math article
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Let <math>S = \{\{x,y\}\ |\ x \in X \and y \in Y\}</math>. The vertical bar means "such that", and the epsilon-like symbol <math>\in</math> means "is an element of". So this means that S is the collection of sets<ref>Mathematicians don't like to use the phrase "set of sets", since this can lead to some unpleasant paradoxes. So they say "collection of sets." A "collection" is really just a set.</ref> which are pairs x and y such that x is in X and y is in Y.
Let <math>S = \{\{x,y\}\ |\ x \in X \textrm{ and } y \in Y\}</math>. The vertical bar means "such that", and the epsilon-like symbol <math>\in</math> means "is an element of". So this means that S is the collection of sets<ref>Mathematicians don't like to use the phrase "set of sets", since this can lead to some unpleasant paradoxes. So they say "collection of sets." A "collection" is really just a set.</ref> which are pairs x and y such that x is in X and y is in Y.
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Now try a harder example, based on the same two sets. Let
Now try a harder example, based on the same two sets. Let
:<math>G = \{F\ |\ F \subset (X \cup Y) \and \|F\|=2\}</math>.
:<math>G = \{F\ |\ F \subset (X \cup Y) \textrm{ and } \|F\|=2\}</math>.
Now <math>X \cup Y = \{a,b,c,p,q,r\}</math>, so
Now <math>X \cup Y = \{a,b,c,p,q,r\}</math>, so
:<math>\|X \cup Y\| = 6</math>.
:<math>\|X \cup Y\| = 6</math>.
:<math>{n \choose k} = \frac{n!}{k! (n-k)!}</math>.
:<math>{n \choose k} = \frac{n!}{k! (n-k)!}</math>.
In the case of the set <math>G = \{F\ |\ F \subset \{a,b,c,p,q,r\} \and \|F\|=2\}</math>, the set is
In the case of the set <math>G = \{F\ |\ F \subset \{a,b,c,p,q,r\} \textrm{ and } \|F\|=2\}</math>, the set is
:<math>G = \{\{a,b\}, \{a,c\}, \{a,p\}, \{a,q\}, \{a,r\}, \{b,c\}, \{b,p\}, \{b,q\}, \{b,r\}, \{c,p\}, \{c,q\}, \{c,r\}, \{p,q\}, \{p,r\}, \{q,r\}\}\,</math>.
:<math>G = \{\{a,b\}, \{a,c\}, \{a,p\}, \{a,q\}, \{a,r\}, \{b,c\}, \{b,p\}, \{b,q\}, \{b,r\}, \{c,p\}, \{c,q\}, \{c,r\}, \{p,q\}, \{p,r\}, \{q,r\}\}\,</math>.
Its cardinality is 15.
Its cardinality is 15.