# What does the big cap notation mean?

I'm trying to understand How can an ordered pair be expressed as a set? and I don't know what the big cap/cup notations mean when placed next to an ordered pair: $\bigcap(a,b)$ and $\bigcup(a,b)$.

• It mean union $\cup$ or intersection $\cap$ of sets in $(a,b)$. – Sasha Oct 5 '11 at 18:29
• @Sasha So it only makes sense if $a$ and $b$ are sets. :) – Paul Manta Oct 5 '11 at 18:30
• The cap is intersection, the cup is union. Remember that the ordered pair $(a,b)$ is just shorthand notation for the set $\{ \{ a \} , \{a,b\} \}$. – Ragib Zaman Oct 5 '11 at 18:32
• In the question you linked to $(a,b)$ represented a set of sets. Then $\cup (a,b)$ denoted the union of those sets. – Sasha Oct 5 '11 at 18:33

This is actually answered in the linked question, but for clarification, if by definition $(a,b)=\{\{a\},\{a,b\}\}$ then $$\bigcap(a,b) = \bigcap\{\{a\},\{a,b\}\} = \{a\} \cap \{a,b\} = \{a\}$$ and $$\bigcup(a,b) = \bigcup\{\{a\},\{a,b\}\} = \{a\} \cup \{a,b\} = \{a,b\}.$$