Elementary Set Theory - Quick Review

Mark each of the following sets as valid or invalid

1. 1($) Set of natural numbers.
2. 2($) Set of roots of the equation, x²  + 7x + 6 = 0.
3. 3($) A set consisting of prime minister of Australia, capital of the United States, natural numbers 1–10, Taj Mahal and alphabets a–c.
4. 4($) Set of rich people.
5. 5($) Set of cleaver students.

If a particular element x belongs to set A, we write it as

x ∊ A

If two elements x and y belong to set A, we shall write it as

x and y ∊ A

If an element x does not belong to set A, we write it as

x ∉ A

The set of all positive odd numbers can written as

1. { All positive odd numbers }
2. { 1, 3, 5, 7, …. }
3. { x|x is a positive odd number }

 

A = { x|x is a positive even number }

Given the following set expressions, assign the right meaning to each symbol

1. x, y ∊ A
2. A = { x|x is an odd negative number } here -3 ∊ A, 8 ∉ A
3. A = ∅
4. N = {1, 2, 3, 4, ...}, n(N) = ℵ₀
5. R = {points in a plane}, n(R) = c
6. x ∊ A ⇔ x ∊ B 

1. 1($) capital letter A
2. 2($) small letter x
3. 3($) ∊
4. 4($) ∉
5. 5($) { } (curly brackets)
6. 6($) | (pipe)
7. 7($) , (comma) between x and y
8. 8($) ∅
9. 9($) ℵ₀ (aleph-null)
10. 10($) c
11. 11($) ⇔

A set that contains only one element is called __________

A set that contains only one element is called __________

 

Let A = { x|x + 5 = 5 }

set A = 1($)

Assign the right term for each expression

1. A = ∅ 1($)
2. A = {∅} 2($)

If a set A contains finite number of elements n, we denote the cardinal number of set A by n(A).  In other words, n(A) stands for the number of elements in a finite set.

What are the cardinal number for each of the following sets?

1. A = {1, 3, 4, 5, 6, 7} 1($)
2. B = {a, b, c, d, e, f, g, h} 2($)
3. C = {2, 4, 6, 8, 10} 3($)

The cardinal number of 1($) infinite set, N = {1, 2, 3, 4, ...} (or any other set that is equivalent to N), is denoted by the symbol ℵ₀ (read aleph-null). The symbol “c” is used to denote the cardinal number of an 2($) infinite set, like the set of all real numbers or the set of points in an open interval or the set of points in a plane.

What is the cardinal number of the empty set

n(∅) = 1($)

Please type a number

Given A = {1, 2, 3, 4, 5}, B = {1, 2, 3}, C = {2, 3, 1}, D = {6, 7, 8}.  Mark true or false for the following

1. A ⊆ B 1($)
2. B ⊆ A 2($)
3. C ⊆ B 3($)
4. B ⊆ C 4($)
5. D ⊈ A 5($)

 

B ⊇ A

The set B is called the 1($) of A

 

x ∊ A ⇔ x ∊ B 

The above statement means

Consider A = {1, 2, 3} and B = {1, 2, 4, 5, 6}. Here, 3 ∈ A but 3 ∉ B. Hence, A ⊄ B.

The sets A = {1, 2} and B = {1, 2, 3} are 1($) as A ⊆ B. On the other hand, the sets C = {1, 2, 3} and D = {2, 3, 4} are 2($).

A = {∅, {1, 2}, {3}, {4, 5}}

Let S = {1, 2, 3}, Then P(S) is

 

S = {1, 2, 3}

 

∅ has 1($) elements

 

A ∪ B = {x|x ∊ A 1($) B 2($) both}

 

A = {1, 2, 3}, B = { 2, 3, 4, 5, 6}

 

A ∪ ∅ = A

It is the set of all those elements that belong to both A 1($) B

A ∩ B = {x|x ∊ A 2($) x ∊ B}

Let A = {1, 2, 3}, B = {2, 3, 5, 6}, and C = {5, 7, 8}. 

Complete the following intersection operations with the right values

• A ∩ B = 1($)
• B ∩ C = 2($)
• A ∩ 3($) = ∅

Let A = {1, 2, 3} and B = {5, 6, 7}, then A ∩ B = ∅ 

The two sets are called 1($) sets

If A = {1, 2, 3, 4}, B = {1, 3, 5, 6} THEN

A - B = B - A

Let U be the universal set and A is a proper subset of U, Then

A' = 1($)

Let U the universal set, A a proper subset of U

• (A')^' = U - A' = U - (1($)) = A
• ∅' = 2($) - ∅ = 3($)
• U' = 4($) = ∅

Let N = {1, 2, 3, 4, ...} = U and A' = {2, 4, 6, 8, ...}, then A =