Question

Let $C$ be the set all complex number. Let $P=\{zϵC:|z|=1\}$ and $Q=\{zϵC:z^2=1\}$. Then

• A. $P=Q$
• B. $P$ is a proper subset of $Q$
• C. $P\cap Q$ is the empty set
• D. $Q$ is a proper subset of $P$

Answer 1

The correct option is D $Q$ is a proper subset of $P$

from given information for $P$, $|z|=1$

Let $z=x+iy$, then $|z|=\sqrt{x^2+y^2}$

Now if $|z|=1=\sqrt{x^2+y^2}$ or $x^2+y^2=1$ ( which is a circle )

Hence $|z|=1$ is a locus of circle on Cartesian plane.

From given information, we know that for $Q$, $z^2=1$

$\to z^2-1=0$ or $(z-1)(z+1)=0$

$\to z=\pm 1$, as the complex part of the $z$ is zero, so we can write,

$\to z_1=1$

$\to z_2=-1$

hence we can see that $Q$ contains only two points $z_1$ and $z_2$ while $P$ contains the locus of points with equal distance from the origin or center $(0,0)$.

As $P$ contains a whole circle $|z|=1$, whereas $Q$ contains only two points on that circle $z_1=1$ and $z_2=-1$., that's why $Q$ is a proper subset of $P$.

Correct answer is $D$.