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6.7 Definition of Complement of Sets What are you going to learn? To define complement of set To determine complement of set To determine the difference between two sets To show complement of set in a Venn diagram To show the difference between two sets in a Venn diagram EXAMPLE 1 Given U a set of all subjects in your school, or U = {PPKn, Bahasa Indonesia, Matematika, Ekonomi, PKK, IPA, IPS, Bahasa Inggris, Pendidikan Jasmani, Kesenian}. If the set M = {IPA, Matematika} and U is its universal set, then which subjects are elements of the set U, but not elements of the set M? Key Terms: • • complement Set the difference between two sets EXAMPLE 2 Given U is the set of all Latin alphabets written as U = {All letters in Latin alphabet}. If the set V = {vowel in Latin alphabet} and U is its universal set, then which letters are elements of the set U but not in set V? In the Example 1, PPKn, Bahasa Indonesia, Bahasa Inggris, Ekonomi, PKK, IPS, Pendidikan Jasmani, and Kesenian are elements of the universal set U, but not in M. In Example 2, consonants like b and n are elements of the universal set U but not in set V. The subjects that are not elements of the set M and letters that are not elements of the set V, are subsets of the universal set U. Mathematics for Junior High School Year 7 / 2 6 7 Such subsets are called complement sets. The complement set of the set M is written as M’ and read as “the complement of set M” or “complement M”. The complement of the set V is written as V’ and read as “complement V” Given set A and its universal set is U. Then the complement of A, or A’, is: A’ = {x x ∈ U and x ∉ A} The Venn Diagram of Complement of Set Notice again the set of subjects and letters in Latin alphabet in Example 1 and 2. The solution of each example is: EXAMPLE 3 a. U = {PPKn, Bhs Indonesia, Matematika, Ekonomi, PKK, IPA, IPS, Bhs Inggris, Penjas, Kesenian} M = { IPA, Matematika } M’ = {PPKn, Bhs Indonesia, Bhs Inggris, Ekonomi, PKK, IPS, Penjas, Kesenian} The Venn diagram is as follows: U • IPS • Ekonomi M • Penjas • PPKn • PKK • B. Indon. IPA• IPA MAT • Mat. • B. Inggris • Kesenian The shaded area is M’ 268 / Student’s Book - Sets b. S = {a, b, c, d, ..., x, y, z} V = { a, e, i, o, u} V’ = {b, d, f, g, h, j, k, l, m, n, p, q, r, s, t, v, w, , y, z} The Venn diagram is as follows: U V consonants vowels The shaded area is V’ To understand the relation between a certain set, its complement and its universal set, copy and then complete the following table. EXAMPLE 4 Intersection of the set and its complement ........... Union of the set and its complemen t .............. Universal Set Set Complement Cardinality U={subjects in SMP} U={letters in Latin alphabet} U={3,4,7, 10,12,15,28} ................... .................... M={IPA, Matematika} V={vowels} K={4,12, 28} ................... ................... ................. n(E)+n(E’)= ................ ............... ................. .......... .............. ................. ................. ................. .......... ........... ........... ................ .............. ............... ................... ................. .................. Mathematics for Junior High School Year 7 / 2 6 9 Based on the above activity, it can be concluded that: The relation between sets, its complement, and its universal set (1) E ∩ E’ = ∅ (2) E ∪ E’ = U (3) n(E)+n(E’)= n(U) The union or the intersection of two sets also has a complement. EXAMPLE 5 Given U = the set of the first 40 natural numbers, A = the set of the first 6 squared natural numbers, B = the set of the first 6 natural numbers of multiple of four find (A ∩ B)’. Solution: Because U = {1, 2, 3, ..., 38, 39, 40} A = { 1, 4, 9, 16, 25, 36 } and B = { 4, 8, 12, 16, 20, 24 } then A ∩ B = { 4, 16} and (A ∩ B)’= {1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40} The Venn diagram is as follows. U A B The shaded area shows (A ∩ B)’ 270 / Student’s Book - Sets C The Difference between Two Sets At the beginning of this subchapter, we have described the complement set of the universal set U. Now we are going to learn about the complement of a set in relation to another set. EXAMPLE 6 Look at the following sets A and B: A = {1, 2, 3, 4, 5} and B = {2, 5, 7, 11}. In those two sets, find the elements of B that are not included in A. Using the definition of complement, the complement of A to B is the set of elements in B but not in A, i.e. {7, 11}. The complement B to A is the set of elements in A, but not in B, i.e. {1, 3, 4}. The complement B to A, written as A – B, is read as “present in A but not in B”. The complement A to B, written as B – A, is read as “present in B, but not in A.” For the set above; (i) B – A = {7, 11} (ii) A – B = {1, 3, 4} The notation of the difference between two sets is written as follows. Mathematics for Junior High School Year 7 / 2 7 1 Given set A and B. Then the difference is: A - B= {x x ∈ A and x ∉ B} B - A= {x x ∈ B and x ∉ A} EXAMPLE 7 Given P = {1, 3, 5} and Q = {2, 4, 6}. Because P ∩ Q = ∅, then P – Q = P = {1, 3, 5} and Q – P = {2, 4, 6}. Study the diagrams below. U 1 3 5 P 2 4 6 P–Q=P Q U 1 3 5 P 2 4 6 Q Q–P=Q 272 / Student’s Book - Sets 1. Show that if A is any set and U is its universal set, then: a. ∅’ = U b. U’ = ∅ c. (A’)’ = A 2. Given U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} A = { 1, 2, 3, 5, 7 } B = { 4, 5, 6, 7, 9 } Listing the elements, determine: a. (A ∪ B)’ b. (A ∩ B)’ c. B - C d. Draw the Venn diagram. 3. Given U = { x ¦ x ≥ 5, x is natural number } B = { x ¦ 5 < x < 8, x is natural number } C = { x ¦ 5 ≤ x ≤ 10, x is natural number } Listing the elements, determine: a. (B ∪ C)’ b. (B ∩ C)’ c. B - C d. Draw the Venn diagram. 4. Given U = the set of squared natural numbers less than 30. L = the set of multiple of 5 less than 30. E = the set of multiple of 6 less than 35. Listing the elements, determine: a. L ∩ E b. E ∪ L c. E – L d. L - E e. Draw the Venn diagram. Mathematics for Junior High School Year 7 / 2 7 3 5. Among 60 people, there are 20 people subscribing to magazines, 35 people subscribing to newspapers, and 5 people subscribing to both. a. Draw the Venn diagram showing the data, letting M = the set of people subscribing to magazines, and K = the set of people subscribing to newspapers. b. How many people neither subscribe to magazines nor to newspapers? c. How many people subscribe not only newspaper? d. How many people subscribe not only magazine? e. How many people subscribe to newspaper but not magazine? 6. Among 50 students, there are 20 students who play tennis, 33 students who play basketball, and 8 students who play both. a. Draw the Venn diagram showing the data. b. How many students who neither play tennis nor basketball? c. How many students do not play tennis? d. How many students do not play basketball? e. How many students do not play tennis but play basketball? 7. Among 50 people shopping in a market, there are 25 people buy apples, 23 people buy bananas, and 8 people buy both fruits. a. Draw the Venn diagram showing the data. b. How many people neither buy apples nor bananas? c. How many people do not buy apples? 8. Given: U = set of students who like specific foods A = set of students who like soup B = set of students who like meatballs G = set of students who like gado-gado. The Venn diagram is as follows. 274 / Student’s Book - Sets (number in the Venn diagram is showing the number of students) U 15 A 5 12 18 11 17 9 B 13 G Determine the number of students who: a. like neither meatballs nor soup. b. like neither meatballs nor gado-gado. c. do not like meatballs. d. don’t like gado-gado. e. like meatballs but not gado-gado. Mathematics for Junior High School Year 7 / 2 7 5 276 / Student’s Book - Sets