2.8 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICSAlgebra
(i) φ has only one subset { φ }
(ii) φ ≠ { φ } but φ ∈ { φ }; {2} ≠ 2.- The complement of the complement of a set A is the set A itself, i.e., (A ́) ́ = A.
SOLVED EXAMPLES
Example 1: Rewrite the following examples using set notation :
(i) First ten even natural numbers.
(ii) Set of days of a week.
(iii) Set of months in a year which have 30 days.
(iv) The numbers 3, 6, 9, 12, 15.
(v) The letters m, a, t, h, e, m, a, t, i, c, s.
Solution:
(i) A = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20} (Tabular method)
= { x : x is an even integer and 2 ≤ x ≤ 20} (Select method)
(ii) A = {Sunday, Monday, ....., Saturday} (Tabular method)
= { x : x is a day in a week} (Selector method)
(iii) A = {April, June, September, November} (Tabular method)
= {x : x is a month of 30 days} (Selector)
(iv) A = { x : x is a positive number multiple of 3 and 3 ≤ x ≤ 15}
(v) A = { x : x is a letter in the word mathematics}.
Example 2 : Write the following set in roster form.
(i) A = {x : x is an integer, –3 ≤ x < 7}
(ii) B = {x : x ix an integer, 4 < x ≤ 12}
Solution:
(i) A = { –3, –2, –1, 0, 1, 2, 3, 4, 5, 6}
(ii) B = { 6, 8, 10, 12}
Example 3 : Represent the following sets in a selector method :
(i) all numbers less than 15
(ii) all even numbers
Solution:
Taking R to be the set of all real numbers in every case :
(i) {x : x ∈ R and x < 15}
(ii) {x : x ∈ R and x is a multiple of 2}
Example 4 : State :
(i) Is the set A = {x : x < x} a null?
(ii) Is the set B = {x : x + 4 = 4} a null?
(iii) Is the set C = { x : x is a positive number less than zero} a null?
Solution:
(i) Null, as there exists no number less than itself.
(ii) Not null, the set has an element zero.