Q. 1.1 taken from book titled: First-Semester Abstract Algebra: A Structural Approach, by: Jessica K. Sklar.
State if each of the below objects stated below is a well-defined set or not.
$\{z\in \mathbb{C}, |z|=1\}$.
$\{x \; ε \;R+, \;\text{is sufficiently small}\;\}$.
$\{q \;ε \;Q : q \;\text{can be written with denominator}\; 4.\}$
$\{n ε Z, n^2 < 0.\}$
Have attempted, and request vetting of the same below.
- Yes, it is well defined as though infinite number of such complex numbers might exist, but each satisfies the fact that if $z= x+iy$, then $\sqrt{x^2+y^2}=1$.
- Not well defined, unless state the numbers in a given range from some real number.
Say, set of numbers in range $0.01$ from $2.345$. Then, all real numbers in range from $2.335$ to $2.355$.
- All rationals are defined in form ($\frac pq$, with $p,q$ as integers) with no common factor between $p,q$. So, the set is of all rationals s.t. $q=4$, hence $p$ is not expressible as a multiple of $4$.Hence, this set is well-defined.
- Such set is empty, as square of any integer (positive, zero or negative) is always either $0$ or positive. Hence, this set is well-defined.