Section 2.3 Using Venn Diagrams to Study Set Operations Math in Our World

Learning Objectives  Illustrate set statements involving two sets with Venn diagrams.  Illustrate set statements involving three sets with Venn diagrams.  Use De Morgan’s laws.  Use Venn diagrams to decide if two sets are equal.  Use the formula for the cardinality of a union.

Illustrating a Set Statement with a Venn Diagram Step 1 Draw a diagram for the sets, with Roman numerals in each region. Step 2 Using the Roman numerals, list the regions for each set. Step 3 Find the set of numerals that correspond to the set given in the set statement. Step 4 Shade the area corresponding to the set of numerals found in step 3. Region I represents the elements in set A that are not in set B. Region II represents the elements in both sets A and B. Region III represents the elements in set B that are not in set A. Region IV represents the elements in the universal set that are in neither set A nor set B. A IIIIII U B IV

EXAMPLE 1 Drawing a Venn Diagram Draw a Venn diagram to illustrate the set (A  B). SOLUTION Step 1 Draw the diagram and label each area. Step 2 From the diagram, list the regions that make up each set. U = {I, II, II, IV}A = {I, II}B = {II, III} Step 3 Using the sets in step 2, find (A  B). First, all of I, II, and III are in either A or B, so A  B = {I, II, II}. The complement is (A  B) = {IV}. Step 4 Shade region IV to illustrate (A  B). IIIIII B IV A U

EXAMPLE 2 Drawing a Venn Diagram Draw a Venn diagram to illustrate the set A  B. SOLUTION Step 1 Draw the diagram and label each area. (same as Example 1) Step 2 From the diagram, list the regions that make up each set. U = {I, II, II, IV}A = {I, II}B = {II, III} Step 3 Using the sets in step 2, find A  B. First, B = {I, IV}. Of these two regions, I is also in set A, so A  B = {I}. Step 4 Shade region I to illustrate A  B. IV U B IIIIII A

Venn Diagrams with Three Sets Region I represents the elements in set A but not in set B or set C. Region II represents the elements in set A and set B but not in set C. Region III represents the elements in set B but not in set A or set C. Region IV represents the elements in sets A and C but not in set B. Region V represents the elements in sets A, B, and C. Region VI represents the elements in sets B and C but not in set A. Region VII represents the elements in set C but not in set A or set B. Region VIII represents the elements in the universal set U, but not in set A, B, or C.

EXAMPLE 3 Drawing a Venn Diagram with Three Sets Draw a Venn diagram to illustrate the set A  (B  C). SOLUTION Step 1 Draw the diagram and label each area. Step 2 From the diagram, list the regions that make up each set. U = {I, II, III, IV, V, VI, VII, VIII}A = {I, II, IV, V} B = {II, III, V, VI}C = {IV, V, VI, VII} Step 3 Using the sets in step 2, first B  C = {V, VI}, so (B  C) = {I, II, III, IV, VII, VIII}. Of these regions, I, II, IV is also in set A, so A  (B  C) = {I, II, IV}. Step 4 Shade regions I, II, IV to illustrate A  (B  C).

De Morgan’s Laws For any two sets A and B, (A  B) = A  B (A  B) = A  B The first law states that the complement of the union of two sets will always be equal to the intersection of the complements of each set. The second law states that the complement of the intersection of two sets will equal the union of the complements of the sets.

EXAMPLE 4Using De Morgan’s Laws If U = {a, b, c, d, e, f, g, h}, A = {a, c, e, g}, and B = {b, c, d, e}, find (A  B) and A  B. SOLUTION A  B = {a, b, c, d, e, g}, so (A  B) = {f, h} A = {b, d, f, h} and B = {a, f, g, h}, so A  B = {f, h} Notice that (A  B) and A  B are equal, illustrating the first of De Morgan’s laws.

EXAMPLE 5Understanding Subset Notation If U = {10, 11, 12, 13, 14, 15, 16}, A = {10, 11, 12, 13}, and B = {12, 13, 14, 15}, find (A  B) and A  B. SOLUTION A  B = {12, 13}, so (A  B) = {10, 11, 14, 15, 16} A = {14, 15, 16} and B = {10, 11, 16}, so A  B = {10, 11, 14, 15, 16} Notice that (A  B) and A  B are equal, illustrating the second of De Morgan’s laws.

EXAMPLE 6 Using a Venn Diagram to Show Equality of Sets Use Venn diagrams to show that (A  B) = A  B. SOLUTION Start by drawing the Venn diagram for (A  B). This was done in Example 1, giving a result of… Next draw the Venn diagram for A  B. Step 1 Draw a second Venn diagram with two sets. Step 2 From the diagram, list the regions that make up each set. U = {I, II, II, IV}A = {I, II}B = {II, III} Step 3 A = {III, IV} and B = {I, IV}, so A  B = {IV}. Step 4 Shade region IV to illustrate A  B. IIIIII B IV A U IIIIII B IV A U

EXAMPLE 7 Using Venn Diagrams to Decide If Two Sets Are Equal Determine if the two sets are equal by using Venn diagrams: (A  B)  C and (A  C)  (B  C)

EXAMPLE 7 Using Venn Diagrams to Decide If Two Sets Are Equal SOLUTION Start with (A  B)  C. The set A  B = {I, II, III, IV, V, VI}. Of these, IV, V, and VI are also in C, so (A  B)  C = {IV, V, and VI}. Now let’s examine (A  C)  (B  C). The set A  C = {IV, V}, and the set B  C = {V, VI}. (A  C)  (B  C) = {IV, V, VI}. Since the shaded areas are the same, the two sets are equal.

Cardinality of a Union If n(A) represents the cardinal number of set A, then for any two finite sets A and B, n(A  B) = n(A) + n(B) – n(A  B). In words, the formula says that to find the number of elements in the union of A and B, you add the number of elements in A and B and then subtract the number of elements in the intersection of A and B.

EXAMPLE 8 Using the Formula for Cardinality of a Union In a survey of 100 randomly selected freshmen walking across campus, it turns out that 42 are taking a math class, 51 are taking an English class, and 12 are taking both. How many students are taking either a math class or an English class? SOLUTION If we call the set of students taking a math class A and the set of students taking an English class B, we’re asked to find n(A  B). We’re told that n(A) = 42, n(B) = 51, and n(A  B) = 12. So… n(A  B) = n(A) + n(B) – n(A  B) = – 12 = 81