Question

# Express the following in set-builder notation: a)The set A of natural numbers divisible by 3. b)The set B of pairs (a,b) of real numbers such that a + b is an integer. c)The open interval C = (—2,2). d)The set D of 20 element subsets of N.

Discrete math
Express the following in set-builder notation:
a)The set A of natural numbers divisible by 3.
b)The set B of pairs (a,b) of real numbers such that a + b is an integer.
c)The open interval C = (—2,2).
d)The set D of 20 element subsets of N.

2021-08-03
The set of all natural numbers is denoted by N, the set of all integers is denoted by Z and the set of all real numbers is denoted by R.
(a) Given that set A is the set of all natural numbers divisible by 3.
A number is divisible by 3 if it is a multiple of 3.
So, every element of A will be of the form 3n where n is a natural number.
Hence, the set-builder notation of A is $$\displaystyle{A}={\left\lbrace{3}{n}:{n}\in{N}\right\rbrace}$$.
(b) Given that set B is the set of all pairs (a,b) of real numbers such that a+b is an integer.
So, every element of B will be of the form (a,b) where $$\displaystyle{a},{b}\in{R}{\quad\text{and}\quad}{a}+{b}\in{Z}$$.
Hence, the set-builder notation of B is $$\displaystyle{B}={\left\lbrace{\left({a},{b}\right)}:{a},{b}\in{R}{\quad\text{and}\quad}{a}+{b}\in{Z}\right\rbrace}$$.
(c) Given that set C is the open interval (−2,2).
Every element of the open interval (−2,2) is a real number such that it lies between -2 and 2 and never equal to -2 or 2.
Hence, the set-builder notation of C is $$\displaystyle{C}={\left\lbrace{x}:{x}\in{R}\right.}$$ and −2 < x < 2}.
(d) Given that set D is the set of all 20 element subsets of N.
So, every element of D is a subset of N and the cardinality of each set in D is 20.
Hence, the set-builder notation of D is $$\displaystyle{D}={\left\lbrace{X}:{X}\subseteq{N}{\quad\text{and}\quad}{\left|{X}\right|}={20}\right\rbrace}$$