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Chapter 2 Sets

WHAT YOU WILL LEARN • Venn diagrams • Set operations such as complement, intersection, union, difference and Cartesian product • Equality of sets • Application of sets • Infinite sets

Venn Diagrams and Set Operations Section 3 Venn Diagrams and Set Operations

Venn Diagrams A Venn diagram is a technique used for picturing set relationships. A rectangle usually represents the universal set, U. The items inside the rectangle may be divided into subsets of U and are represented by circles.

Disjoint Sets Two sets which have no elements in common are said to be disjoint. The intersection of disjoint sets is the empty set. Disjoint sets A and B are drawn in this figure. There are no elements in common since there is no overlap- ping area between the two circles.

Overlapping Sets For sets A and B drawn in this figure, notice the overlapping area shared by the two circles. This area represents the elements that are in the intersection of set A and set B.

Complement of a Set The set known as the complement contains all the elements of the universal set which are not listed in the given subset. Symbol: A´

Intersection The intersection of two given sets contains only those elements common to both of those sets. Symbol:

Union The union of two given sets contains all of the elements for both of those sets. The union “unites”, that is, it brings together everything into one set. Symbol:

Subsets When every element of B is also an element of A. Circle B is completely inside circle A.

Equal Sets When set A is equal to set B, all the elements of A are elements of B, and all the elements of B are elements of A. Both sets are drawn as one circle.

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. n(A  B) = n(A) + n(B) – n(A  B)

Difference of Two Sets The difference of two sets A and B symbolized by 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. U A B I II III IV

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. 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.

Section 4 Venn Diagrams with Three Sets And Verification of Equality of Sets

General Procedure for Constructing Venn Diagrams with Three Sets Determine the elements that are common to all three sets and place in region V, A  B  C.

General Procedure for Constructing Venn Diagrams with Three Sets (continued) Determine the elements for region II. Find the elements in A  B. The elements in this set belong in regions II and V. Place the elements in the set A  B that are not listed in region V in region II. The elements in regions IV and VI are found in a similar manner.

General Procedure for Constructing Venn Diagrams with Three Sets (continued) Determine the elements to be placed in region I by determining the elements in set A that are not in regions II, IV, and V. The elements in regions III and VII are found in a similar manner.

General Procedure for Constructing Venn Diagrams with Three Sets (continued) Determine the elements to be placed in region VIII by finding the elements in the universal set that are not in regions I through VII. U I II III A V B IV VI VII C VIII

Example: Constructing a Venn diagram for Three Sets Construct a Venn diagram illustrating the following sets. U = {1, 2, 3, 4, 5, 6, 7, 8} A = {1, 2, 5, 8} B = {2, 4, 5} C = {1, 3, 5, 8} Solution: Find the intersection of all three sets and place in region V, A  B  C = {5}.

Example: Constructing a Venn diagram for Three Sets (continued) Determine the intersection of sets A and B. A  B = {2, 5} Element 5 has already been placed in region V, so 2 must be placed in region II. Now determine the numbers that go into region IV. A  C = {1, 5, 8} Since 5 has been placed in region V, place 1 and 8 in region IV.

Example: Constructing a Venn diagram for Three Sets (continued) Now determine the numbers that go in region VI. B  C = {5} There are no new numbers to be placed in region VI. Since all numbers in set A have been placed, there are no numbers in region I. The same procedures using set B completes region III, in which we must write 4. Using set C completes region VII, in which we must write 3.

Example: Constructing a Venn diagram for Three Sets (continued) Now place the remaining elements in U (6 and 7) in region VIII. The Venn diagram is then completed. U A B C V I III VII VI IV VIII II 2 4 5 1,8 3 6 7

Verification of Equality of Sets To verify set statements are equal for any two sets selected, we use deductive reasoning with Venn Diagrams. If both statements represent the same regions of the Venn Diagram, then the statements are true for all sets A and B.

Example: Equality of Sets Determine whether (A  B)´ = A´ B´ for all sets A and B.

Solution Find A´ B´. B A U I II III IV Find (A  B)´. Draw a Venn diagram with two sets A and B and label each region. Find (A  B)´. Find A´ B´. Set Regions A I, II B II, III A  B II (A  B)´ I, III, IV Set Regions A´ III, IV B´ I, IV A´ B´ I, III, IV

Solution Set Regions A I, II B II, III A  B II (A B)´ I, III, IV Set Regions A´ III, IV B´ I, IV A´ B´ I, III, IV Both statements are represented by the same regions, I, III, and IV, of the Venn diagram. Thus, (A  B)´ = A´ B´ for all sets A and B.

De Morgan’s Laws A pair of related theorems known as De Morgan’s laws make it possible to change statements and formulas into more convenient forms. (A B)´ = A´ B´ (A B)´ = A´ B´