Exercise 1.1:
1. List the elements of the following sets:
(a) {x:x” is an integer and ” x^2=9}
Answer:
x=±3
{-3,3}
(b) {x:x” is a positive integer less than ” 5}
Answer:
Positive integers less than 5: 1, 2, 3, 4
{1,2,3,4}
(c) {x:x” is an even natural number divisible by ” 5}
Answer:
Even numbers divisible by 5 are multiples of 10: 10, 20, 30, …
{10,20,30,40,…}
(d) {x:x∈N” and ” x<-1}
Answer:
No natural number is less than -1 (natural numbers start from 1).
ϕ" (empty set)"
2. Determine which elements of the set
A={-5,-√3,-1/2,0,2/5,π,13.4,1/3,19/2}, are:
(a) Natural numbers
Answer:
Natural numbers are positive integers (1, 2, 3, ...). None of the elements are positive integers.
{}" or " ϕ
(b) Whole numbers
Answer:
Whole numbers are non-negative integers (0, 1, 2, 3, ...). Only 0 fits.
{0}
(c) Integers
Answer:
Integers are whole numbers and their negatives. From the set: -5, 0 are integers.
{-5,0}
(d) Rational numbers
Answer:
Rational numbers can be expressed as p/q where p,q are integers, q≠0.
From the set: -5,-1/2,0,2/5,13.4(=134/10),1/3,19/2
(Note: -√3 and π are irrational)
{-5,-1/2,0,2/5,13.4,1/3,19/2}
(e) Real numbers
Answer:
All elements are real numbers (all numbers in the set are real).
{-5,-√3,-1/2,0,2/5,π,13.4,1/3,19/2}
3. Write the following sets in roster form:
(a) {x:x" is a two digit number and the sum of the digits is " 5}
Answer:
Two-digit numbers where digits sum to 5: 14, 23, 32, 41, 50
{14,23,32,41,50}
(b) {x:x" is an integer and "|x|≥9}
Answer:
Integers with absolute value ≥ 9: x≤-9 or x≥9
{…,-11,-10,-9,9,10,11,…}
(c) {x:x" is a letter of the word “SWEET”"}
Answer:
Unique letters in "SWEET": S, W, E, T
{S,W,E,T}
(d) {x:x=(n+1)/n," where " n" is a natural number and " n<6}
Answer:
n = 1, 2, 3, 4, 5
n=1: 2/1 = 2
n=2: 3/2 = 1.5
n=3: 4/3
n=4: 5/4
n=5: 6/5
{2,3/2,4/3,5/4,6/5}
(e) {x:x" is a composite number"}
Answer:
Composite numbers are positive integers with more than two factors: 4, 6, 8, 9, 10, ...
{4,6,8,9,10,12,14,15,…}
4. Write the following sets in set-builder form.
(i) {2,4,6,8,…}
Answer:
{x:x" is an even natural number"} "or" {x:x=2n,n∈N}
(ii) {3,6,9,12,15}
Answer:
{x:x" is a multiple of 3 and " 3≤x≤15} "or" {x:x=3n,n∈N,1≤n≤5}
(iii) {1,4,9,16,…}
Answer:
{x:x=n^2,n∈N}
(iv) {8,9,10,11,…}
Answer:
{x:x" is an integer and " x≥8}
(v) {1,2,3,6}
Answer:
{x:x" is a positive divisor of " 6}
Can two different sets have the same roster form?
Answer:
No. Roster form uniquely determines a set. If two sets have the same roster form, they are the same set.
5. Which of the following pairs of sets are equal.
(i) {D,E,C,E,N,T} and {C,E,N,T,D}
Answer:
First set has duplicate E, but duplicates don't matter: {D,E,C,N,T}. Second set is {C,E,N,T,D}.
These are equal.
"Equal"
(ii) {a,b,π,√2} and {a,π,√2,b}
Answer:
Same elements, just different order.
"Equal"
(iii) {x:x" is zero of the polynomial " x^2} and {x:x" is the root of the equation, " x^2=0}
Answer:
Both have only the root 0. First set: zero of x^2 is 0. Second set: root of x^2=0 is 0.
"Equal"
(iv) {x:x" has numerical value less than or equal to " 1} and {x:x" is the root of the equation, " x^2-1=0}
Answer:
First set: |x|≤1⇒-1≤x≤1, infinite set.
Second set: roots of x^2-1=0 are -1,1, finite set of 2 elements.
"Not equal"
(v) {5,10,15,20} and {5,10,15,20,…}
Answer:
First is finite set of 4 elements. Second is infinite set (multiples of 5).
"Not equal"
(vi) ϕ and {ϕ}
Answer:
ϕ is the empty set (no elements). {ϕ} is a set containing one element, which is the empty set itself.
"Not equal"
6. State which of the following sets are finite or infinite.
(i) {x:x∈Z" and "(x-1)(x+2)(x-3)=0}
Answer:
Solutions: x=1,-2,3. Only 3 elements.
"Finite"
(ii) {x:x" and " 2" are coprime"}
Answer:
Coprime means gcd(x, 2) = 1. All odd integers satisfy this (and also x=1, -1, -3, ...). Infinite many.
"Infinite"
(iii) {x:x" is a rational number between " 3" and " 4}
Answer:
There are infinitely many rational numbers between 3 and 4 (e.g., 3.1, 3.01, 3.001, ...).
"Infinite"
(iv) {x:x" is an integer and "|x|≥5}
Answer:
Integers with absolute value ≥ 5: x≤-5 or x≥5, infinite many.
"Infinite"