Section 2.3B Venn Diagrams and Set Operations

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 }

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.

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

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}

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?

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) = 25 + 14 – 4 = 35

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.

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 }

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´

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}

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}

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}

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}

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}

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.

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.

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

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)}

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)}

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)}

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)}

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}

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