Question

Let \(S_1, S_2\) and \(S_3\) be three sets defined as

\(S_1=\{z\in\mathbb{C} : |z-1|\le\sqrt{2}\}\)

\(S_2=\{z\in\mathbb{C} : \mathrm{Re}((1-i)z)\ge 1\}\)

\(S_3=\{z\in\mathbb{C} : \mathrm{Im}(z)\le 1\}\)

Then the set \(S_1\cap S_2\cap S_3\) :

• A. has exactly three elements
• B. is a singleton
• C. has infinitely many elements
• D. has exactly two elements

Hint:

Put \(z=x+iy\): \(S_1\) is a disc, \(S_2\) is the half-plane \(x+y\ge 1\) and \(S_3\) is \(y\le 1\). Check whether the intersection region has non-empty interior.

Answer 1

The Correct Option is C

Let \(z=x+iy\).
\(S_1\): \((x-1)^2+y^2\le 2\), a closed disc with centre (1, 0) and radius \(\sqrt{2}\).
\(S_2\): \((1-i)(x+iy)=(x+y)+i(y-x)\), so \(\mathrm{Re}=x+y\ge 1\), a closed half-plane whose boundary line \(x+y=1\) passes through the centre (1, 0) of the disc.
\(S_3\): \(y\le 1\), the half-plane below the line \(y=1\).
The half-disc given by \(S_1\cap S_2\) contains the interior point (1.5, 0.2): \(x+y=1.7\ge 1\), \(y\le 1\), and \((0.5)^2+(0.2)^2\lt 2\). An interior point means a whole neighbourhood of points belongs to the intersection.
Hence the intersection has infinitely many elements.
Answer: (3) has infinitely many elements