Section 1.2 Operations on sets.pdf

Math 241: Discrete mathematics Section 1.2: Operations on Sets Hiyam Al-Bataineh Jordan University of Science and technology S02  The union and subset Operations Let A and B be two sets , then The union of A and B to be all elements belong to A or B. The union is denoted by 𝐴...

Math 241: Discrete mathematics Section 1.2: Operations on Sets Hiyam Al-Bataineh Jordan University of Science and technology S02  The union and subset Operations Let A and B be two sets , then The union of A and B to be all elements belong to A or B. The union is denoted by 𝐴 ∪ 𝐵 In other words 𝑥 ∈ 𝐴 ∪ 𝐵 if either 𝑥 ∈ 𝐴 or 𝑥 ∈ 𝐵. The intersection of A and B to be all elements belong to both A and B. The intersection is denoted by 𝐴 ∩ 𝐵 In other words 𝑥 ∈ 𝐴 ∩ 𝐵 if 𝑥 ∈ 𝐴 and 𝑥 ∈ 𝐵.  Examples Let 𝐴 = {1, 2, 3, 4, 5, 6}, 𝐵 = {2, 4, 6, 8, 10} Then 𝐴 ∪ 𝐵 = 1, 2, 3, 4, 5, 6, 8, 10 And 𝐴 ∩ 𝐵 = 2, 4, 6 In Venn Diagram A B 𝐴∪𝐵 𝐴∩𝐵  Union and Intersection for More Sets 𝐴 ∪ 𝐵 ∪ 𝐶 = {𝑥| 𝑥 ∈ 𝐴 𝑜𝑟 𝑥 ∈ 𝐵 𝑜𝑟 𝑥 ∈ 𝐶} 𝐴 ∩ 𝐵 ∩ 𝐶 = {𝑥| 𝑥 ∈ 𝐴 𝑎𝑛𝑑 𝑥 ∈ 𝐵 𝑎𝑛𝑑 𝑥 ∈ 𝐶} U U U A B A B A B C C C 𝐴∪𝐵 ∪𝐶 𝐴∩𝐵 ∩𝐶  𝑛 𝐴1 ∪ 𝐴2 ∪ ⋯ ∪ 𝐴𝑛 = 𝑥 𝑥 ∈ 𝐴𝑖 𝑓𝑜𝑟 𝑎𝑡 𝑙𝑒𝑎𝑠𝑡 𝑖 = 1, 2, … , 𝑛} = ራ 𝐴𝑖 𝑖=1 𝑛 𝐴1 ∩ 𝐴2 ∩ ⋯ ∩ 𝐴𝑛 = 𝑥 𝑥 ∈ 𝐴𝑖 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖 = 1, 2, … , 𝑛} = ሩ 𝐴𝑖 𝑖=1 Example: Let A = {1, 2, 3, 4}, B = {2,3,4,5,6}, C={3,4,5,6}, Then A ∪ 𝐵 ∪ 𝐶 = 1, 2, 3, 4, 5, 6 , 𝐴 ∩ 𝐵 ∩ 𝐶 = {3, 4}  The Complement of a Set Let A and B be two sets, then the complement of B with respect to A 𝐴 − 𝐵 = 𝑥 𝑥 ∈ 𝐴 𝑎𝑛𝑑 𝑥 ∉ 𝐵} If U is the universal set containing A then 𝑈 − 𝐴 is called the complement of A and denoted by 𝐴ҧ = 𝑥 𝑥 ∉ 𝐴} 𝐴−𝐵 B−𝐴 𝐴ҧ  The symmetric Difference Let A and B be two sets, then the symmetric difference between A and B 𝐴 ⊕ 𝐵 = 𝑥 𝑥 ∈ 𝐴 𝑎𝑛𝑑 𝑥 ∉ 𝐵 𝑜𝑟 𝑥 ∈ 𝐵 𝑎𝑛𝑑 𝑥 ∉ 𝐴} 𝐴 ⊕ 𝐵 = 𝐴 − 𝐵 ∪ 𝐵 − 𝐴 = (𝐴 ∪ 𝐵) − (𝐴 ∩ 𝐵) 𝐴⊕𝐵 Example: Let A = {1, 2, 3,4}, B = {3,4,5,6}, then 𝐴 ⊕ 𝐵 = {1, 2, 5, 6}  Algebraic Properties of Set Operators Commutative Properties: 𝐴 ∪ 𝐵 = 𝐵 ∪ A, 𝐴∩𝐵 =𝐵∩A Associative Properties: 𝐴 ∪ 𝐵 ∪ C = (𝐴 ∪ B) ∪ 𝐶, 𝐴 ∩ 𝐵 ∩ C = (𝐴 ∩ B) ∩ 𝐶 Distributive Properties: 𝐴 ∪ 𝐵 ∩ C = (𝐴 ∪ B) ∩ (𝐴 ∪ 𝐶), 𝐴 ∩ 𝐵 ∪ C = (𝐴 ∩ B) ∪ (𝐴 ∩ 𝐶)  Algebraic Properties of Set Operators Idempotent Properties: 𝐴 ∪ 𝐴 = A, 𝐴∩𝐴=A Properties of the complements: 𝐴=𝐴 𝐴 ∪𝐴=𝑈 𝐴 ∩𝐴= ∅ ∅ =𝑈 𝑈=∅ De Morgan’s Laws: 𝐴 ∪ 𝐵 = 𝐴 ∩ 𝐵 𝐴∩𝐵 = 𝐴∪𝐵  Algebraic Properties of Set Operators Properties of Universal Set 𝐴 ∪ 𝑈 = U, 𝐴∩𝑈 =A Properties of the Empty Set 𝐴 ∪ ∅ = A, 𝐴∩∅=∅  The Cardinality of Union Sets We say that A and B are disjoint if 𝐴 ∩ 𝐵 = ∅ If A and B are disjoint, then 𝐴 ∪ 𝐵 = 𝐴 + |𝐵| A B A B In general 𝐴 ∪ 𝐵 = 𝐴 + 𝐵 − |𝐴 ∩ 𝐵|  The Cardinality of 3 Union Sets 𝐴 ∪ 𝐵 ∪ 𝐶 = 𝐴 + 𝐵 + 𝐶 − 𝐴 ∩ 𝐵 − 𝐴 ∩ 𝐶 − 𝐵 ∩ 𝐶 + |𝐴 ∩ 𝐵 ∩ 𝐶| U A B C  Exercises Exc. 23: In a survey of 260 college students, the following data were obtained 64 had taken a mathematics course. M 94 had taken a computer science course. C 58 had taken a business course. B 28 had taken both a mathematics and a business course. 𝑀 ∩ 𝐵 26 had taken both a mathematics and a computer science course. 𝑀 ∩ 𝐶 22 had taken both a computer science and a business course. 𝐶 ∩ 𝐵 14 had taken all three types of courses. 𝑀 ∩ 𝐶 ∩ 𝐵  Exercises (a) How many students were surveyed who had taken none of the three types of courses? = 260 − |𝑀 ∪ 𝐶 ∪ 𝐵| = 260 – (64 + 94 + 58 − 28 − 26 − 22 + 17) = 260 − 154 = 106 students (a) Of the surveyed students, how many had taken only a computer science course? 𝐶 𝑜𝑛𝑙𝑦 = 𝐶 − M ∩ 𝐶 − All M B 𝐵 ∩ 𝐶 + |𝑀 ∩ 𝐵 ∩ 𝐶| = 94 − 26 − 22 + 14 = 60 students C Exercises Do the following on page 11: 1, 3, 5, 7, 9, 11, 12, 13, 14, 15, 17, 21, 23, 24, 26, 37, 38.