Can someone explain to be why we are taking intersection instead of union? Because taking the union means we are also taking the union of ALL the $y$s in $S$ no?
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Intersection is "in all the sets simultaneously". Now, check again the definition of the set.
Martín-Blas Pérez Pinilla
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And union isn't "all sets"? – Lemon Jan 30 '14 at 08:27
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Element $\in$ union iff $\in$ some of the sets. In all the sets simultaneously (smaller thing) is the intersection. – Martín-Blas Pérez Pinilla Jan 30 '14 at 08:40
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But I can union everything no? – Lemon Jan 31 '14 at 03:35
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You can, but the union isn't your set. – Martín-Blas Pérez Pinilla Jan 31 '14 at 18:18
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Um, why…?Union here is also taking everything – Lemon Feb 03 '14 at 06:43
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Easy example in the plane: what points are in all the lines $y=ax$ simultaneously? – Martín-Blas Pérez Pinilla Feb 03 '14 at 07:09