1. Quiz 3
True or false. If the statement is true, prove it; if not, disprove it or give a counterexample.
(1) ( ) The set
{(x, y) ∈ R2: (x, y) 6= (0, 0)}
is an open subset of R2.
(2) ( ) The point p(1, 1) is an interior point of
{(x, y) ∈ R2: x2+ 4y2≤ 16}.
(3) ( ) The point p(1, 2) is a boundary point of {(x, y) ∈ R2: xy > 1}.
(4) ( ) The point p(0, 0) is an isolated point of {(x, 0) ∈ R2: x ∈ Q}.
Here Q is the set of all rational numbers.
(5) ( ) The set
{(m, n) ∈ R2: m, n ∈ Z, 0 < n < 4}
is a closed set. Here Z is the set of all integers.
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