1. Section 2.3B Venn Diagrams and Set Operations

2. The Meaning of and and or
and is generally interpreted to mean intersection
A ∩ B = { x | x ∈A and x ∈B }
or is generally interpreted to mean union
A ⋃ B = { x | x ∈A or x ∈B }

3. The Relationship Between n(A ⋃ B), n(A), n(B), n(A ∩ B)
To find the number of elements in the union of two sets A and B, we add the number of elements in set A and B and then subtract the number of elements common to both sets.

4. The Number of Elements in A ⋃ B
For any finite sets A and B,
n(A ⋃ B) = n(A) + n(B) – n(A ∩ B)

5. Try This: Find n(A U B)
U = {a, b, c, d, e, f, g, h} A = { a, d, h} B = {b, c, d, e}

6. Example 7: How Many Visitors Speak Spanish or French?
The results of a survey of visitors at the Grand Canyon showed that 25 speak Spanish, 14 speak French, and 4 speak both Spanish and French. How many speak Spanish or French?

7. Example 7: How Many Visitors Speak Spanish or French?
Solution
Set A is visitors who speak Spanish
Set B is visitors who speak French
We need to determine A ⋃ B.
n(A ⋃ B) = n(A) + n(B) – n(A ∩ B)
n(A ⋃ B) = – 4
= 35

8. Difference of Two Sets
The difference of two sets A and B, symbolized A – B, is the set of elements that belong to set A but not to set B.
Region 1 represents the difference of the two sets.

9. Difference of Two Sets
Using set-builder notation, the difference between two sets A and B is indicated by
A – B = { x | x ∈A or x ∉B }

10. Example 9: The Difference of Two Sets
Given
U = {a, b, c, d, e, f, g, h, i, j, k} A = {b, d, e, f, g, h}
B = {a, b, d, h, i}
C = {b, e, g}
Find
a) A – B b) A – C
c) A´– B d) A – C´

11. Example 9: The Difference of Two Sets
Solution
U = {a, b, c, d, e, f, g, h, i, j, k} A = {b, d, e, f, g, h}
B = {a, b, d, h, i}
C = {b, e, g}
a) A – B is the set of elements that are in set A but not in set B.
A – B = {e, f, g}

12. Example 9: The Difference of Two Sets
Solution
U = {a, b, c, d, e, f, g, h, i, j, k} A = {b, d, e, f, g, h}
B = {a, b, d, h, i}
C = {b, e, g}
b) A – C is the set of elements that are in set A but not in set C.
A – C = {d, f, h}

13. Example 9: The Difference of Two Sets
Solution
U = {a, b, c, d, e, f, g, h, i, j, k} A = {b, d, e, f, g, h}
B = {a, b, d, h, i}
c) A´– B is the set of elements that are in set A´ but not in set B.
A´ = {a, c, i, j, k}
A´– B = {c, j, k}

14. Example 9: The Difference of Two Sets
Solution
U = {a, b, c, d, e, f, g, h, i, j, k} A = {b, d, e, f, g, h}
C = {b, e, g}
c) A – C´ is the set of elements that are in set A but not in set C´.
C´ = {a, c, d, f, h, i, j, k}
A – C´ = {b, e, g}

15. Try This: Use the information to find the solutions
U = {a, b, c, d, e, f, g, h} A = { a, d, h} B = {b, c, d, e}

16. Cartesian Product
The Cartesian product of set A and set B, symbolized A × B, and read “A cross B,” is the set of all possible ordered pairs of the form (a, b), where a ∈ A and b ∈ B.

17. Ordered Pairs in a Cartesian Product
Select the first element of set A and form an ordered pair with each element of set B. Then select the second element of set A and form an ordered pair with each element of set B. Continue in this manner until you have used each element in set A.

18. Example 10: The Cartesian Product of Two Sets
Given A = {orange, banana, apple} and B = {1, 2}, determine the following.
a) A × B b) B × A
c) A × A d) B × B

19. Example 5: The Union of Sets
Solution
A = {orange, banana, apple} and
B = {1, 2}
a) A × B = {(orange, 1), (orange, 2), (banana, 1), (banana, 2), (apple, 1), (apple, 2)}

20. Example 5: The Union of Sets
Solution
A = {orange, banana, apple} and
B = {1, 2}
b) B × A = {(1, orange), (1, banana), (1, apple), (2, orange), (2, banana), (2, apple)}

21. Example 5: The Union of Sets
Solution
A = {orange, banana, apple} and
B = {1, 2}
c) A × A = {(orange, orange), (orange, banana), (orange, apple), (banana, orange), (banana, banana), (banana, apple), (apple, orange), (apple, banana), (apple, apple)}

22. Example 5: The Union of Sets
Solution
A = {orange, banana, apple} and
B = {1, 2}
d) B × B = {(1, 1), (1, 2), (2, 1), (2, 2)}

23. Try This: Use the information to find the solutions
U = {a, b, c, d, e, f, g, h} A = { a, d, h} B = {b, c, d, e}

24. Homework
p. 68 # 71 – 84 all, 85 – 110 (x5)