1. Real Numbers Natural Numbers – {1,2,3,4,5,6….}
Whole Numbers – {0,1,2,3,4,5,6…}
Integers – {..-3,-2,-1,0,1,2,3..}
Rational Numbers – {are numbers that CAN be expressed
as a/b, where b≠0.
Irrational Numbers – {are numbers that CANNOT
be expressed as a/b, where b≠0.
π = (and more...)
You cannot write down a simple fraction that equals Pi.
square root of 2, which is

2. A set is a collection of objects, things or symbols which are clearly defined.
Example: B = {2, 4, 6, 8, 10} X = {a, b, c, d, e,f,g}
ELEMENTS are the individual objects in a set.
∈ denotes “is an element of’ or “is a member of” or “belongs to”
Example:
Elements of Set B
Elements of Set X

3. Null Set or Empty Set – is a set with no elements.
It is represented by the symbol { } or Ø .
Example:
The set of squares with 5 sides.
The set of sedan cars with 200 doors.
The set of months with 32 days

4. If every element of a set A is also a member of a set B,
then we say A is a subset of B.
We use the symbol ⊂ to mean “is a subset of” and the symbol ⊄ to mean “is not a subset of”.
Example:
A = {1, 3, 5}, B = {1, 2, 3, 4, 5}
X = {1, 3, 5}, Y = {2, 3, 4, 5, 6}.
X ⊄  Y  because 1 is in X  but not in Y.

5. A universal set is the set of all elements under consideration, denoted by capital U or sometimes capital E.
Example: Given that
U = {5, 6, 7, 8, 9, 10, 11, 12}, list the elements of the following sets.
a) A = {x : x is a factor of 60}
b) B = {x : x is a prime number}

6. The complement of set A, denoted by A’ , is the set of all elements in the universal set that are not in A.
Example:
Let U = {x : x is an integer, –4 ≤ x ≤ 7},
P = {–4, –2, 0, 2, 4, 5, 6}
List the elements of set P ’

7. Example:

8. Practice 1 Universal set U = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 }
A={1,3,5} B={1,2,3, 5,7, 12}
How many elements are there in set A?______
How many elements are there in set B?______
A ⊂ U True or False? ____________________
B ⊂ U True or False? ____________________
A ⊂ B True or False? ____________________
B ⊂ A True or False? ____________________
C = {x : x is a factor of 25} ____________________
D = {x : x is an even number} ____________________

9. In a Venn diagram, the sets are represented by shapes; usually circles or ovals. The elements of a set are labelled within the circle.
Given the set P is the set of even numbers between 15 and 25. Draw and label a Venn diagram to represent the set P and indicate all the elements of set P in the Venn diagram.
Draw and label a Venn diagram to represent the set
R = {Monday, Tuesday, Wednesday}.
Given the set Q = {x : 2x – 3 < 11, x is a positive integer }. Draw and label a Venn diagram to represent the set Q.

10. Let U = {x : x is an integer, -7≤ x ≤ 5}, P = {-6, -4, 0, 1, 5} and
Q ’ = {–3, –2, –1, 3, 4}.
List the elements of set P ’
Find n(Q)
Draw a Venn diagram to display the sets U , P and P ’
Draw a Venn diagram to display the sets U , Q and Q ’

11. Let U = {x : x is an integer, 1≤ x < 10 },
P = { 1, 5, 6} and Q ’ = { 3, 4, 9}.
List the elements of set P ’
Find n(Q)
Draw a Venn diagram to display the sets U , P and P ’
Draw a Venn diagram to display the sets U , Q and Q ’

12. The intersection of two sets X and Y is the set of elements that are common to both set X and set Y.
It is denoted by X ∩ Y and is read ‘X intersection Y’.

13. Draw a Venn diagram to represent the relationship between the sets
X = {1, 2, 5, 6, 7, 9, 10} and
Y = {1, 3, 4, 5, 6, 8, 10}

14. Draw a Venn diagram to represent the relationship between the sets
X = {a, b,c,d,e,f,g,h,i}
and
Y = {b,e,h,j,k,l,m}

15. Draw a Venn diagram to represent the relationship between the sets
X = {1, 6, 9}
and
Y = {1, 3, 5, 6, 8, 9}

16. The intersection of three sets X, Y and Z is the set of elements that are common to sets X, Y and Z.
It is denoted by X ∩ Y ∩ Z

17. Draw a Venn diagram to represent the relationship between the sets
X = {1, 2, 5, 6, 7, 9},
Y = {1, 3, 4, 5, 6, 8} and
Z = {3, 5, 6, 7, 8, 10}
X ∩ Y ∩ Z =
X ∩ Y =
Y ∩ Z =
X ∩ Z =

18. Draw a Venn diagram to represent the relationship between the sets
X = {a,b,c,d,e,f,g},
Y = {a,g,h,j,k} and
Z = {a,b,j,k,l,m,n}
X ∩ Y ∩ Z =
X ∩ Y =
Y ∩ Z =
X ∩ Z =

19. The complement of the set X ∩ Y is the set of elements that are members of the universal set U but not members of X ∩ Y.
It is denoted by (X ∩ Y) ’

20. U ={1,2,3,4,5,6,7,8,9,10,11},
X = {1, 2, 5, 6, 7}
and
Y = {1, 3, 4, 5, 6, 8} .
Draw a Venn diagram to illustrate ( X ∩ Y ) ’
b) Find ( X ∩ Y ) ’

21. Suppose U = set of positive integers less than 10,
X = {2,3,8} and Y = {1,2,7,8} .
Draw a Venn diagram to illustrate ( X ∩ Y ) ’
b) Find ( X ∩ Y ) ’

22. Suppose U = {x : x is an integer, -6≤ x < 5 }
X = {-3,0,1,3} and
Y = {-5,-1,1,3,4} .
Draw a Venn diagram to illustrate ( X ∩ Y ) ’
b) Find ( X ∩ Y ) ’

23. The union of two sets A and B is the set of elements, which are in A or in B or in both. It is denoted by A ∪ B and
is read ‘A union B’

24. Find X ∪ Y and draw a Venn diagram to illustrate X ∪ Y.
X = {1, 2, 6, 7} and
Y = {1, 3, 4, 5, 8}
Find X ∪ Y and draw a Venn diagram to illustrate X ∪ Y.

25. Given U = {-2,-1,3,4,5,9,10}
X = {-2,3,5,9} and
Y = {-1,3,9,10}
Find X ∪ Y and draw a Venn diagram to illustrate X ∪ Y.

26. Given U = U = {x : x is an integer, -6≤ x < 5 }
X = {-3,0,1,3} and
Y = {-5,-1,1,3,4} .
Find X ∪ Y and draw a Venn diagram to
illustrate X ∪ Y.

31. Set Equality
Consider the sets:
P ={Tom, Dick, Harry, John} Q = {Dick, Harry, John, Tom}
Since P and Q contain exactly the same number of members and the memebers are the same, we say that P is equal to Q, and we write P = Q. The order in which the members appear in the set is not important.
R = {2, 4, 6, 8} S = {2, 4, 6, 8, 10}
Since R and S do not contain exactly the same members, we say that R is not equal to S and we write R ≠ S.

32. If every element of a set B is also a member of a set A, then we say B is a subset of A. We use the symbol ⊂ to mean “is a subset of” and the symbol ⊄ to mean “is not a subset of”.
Example: A = {1, 3, 5}, B = {1, 2, 3, 4, 5}
So,  A ⊂  B because every element in A is also in B.
X = {1, 3, 5}, Y = {2, 3, 4, 5, 6}.
X ⊄  Y  because 1 is in X  but not in Y.