• Document: The Mathematics Driving License for Computer Science- CS10410 Venn Diagram, Union, Intersection, Difference, Complement, Disjoint, Subset and Power Set Nitin Naik Department of Computer Sc...
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The Mathematics Driving License for Computer Science- CS10410 Venn Diagram, Union, Intersection, Difference, Complement, Disjoint, Subset and Power Set Nitin Naik Department of Computer Science Venn-Euler Diagram • Venn-Euler diagram or simply Venn diagram is a graphical representation of sets and relation between sets. Venn-Euler Diagram.. • In the Venn diagram, Universal set is represented by a rectangle , other sets are represented by the circles inside the rectangle. • The relation between sets is represented by the way the circles are placed inside the rectangle. Sets Representation in Venn Diagram Sets Representation in Venn Diagram.. • Let us consider a very simple example to represent sets in Venn Diagram: • A = { 1, 2 } • B = { 2, 3 } • U = { 1, 2, 3, 4 } Set Union (A∪B) • The union of two sets is the set of all elements which are in either set. • The union of sets A and B is the set of all elements of A together with the elements of B. • (The A∪B is the set of all x such that x belongs to A or x belongs to B.) Set Union.. • In the previous example: • A = { 1, 2 } • B = { 2, 3 } Union of Multiple Sets • We can also find the union of multiple sets: Set Intersection (A∩B) • The intersection of two sets is the set of all elements which are in both set. • The intersection of sets A and B is the set of all elements of A which are also the elements of B. • (The A∩B is the set of all x such that x belongs to A and x belongs to B.) Set Intersection.. • In the previous example: • A = { 1, 2 } • B = { 2, 3 } • Sometimes there will be no intersection at all. In that case we say the answer is the empty set or the null set. Intersection of Multiple Sets • We can also find the intersection of multiple sets: Set Difference (A−B or A∖B ) • We can extend the concept of subtraction, used in the algebra, to the sets. • If a set B is subtracted from set A, the resulting difference set consists of elements, which are exclusive to set A. • (The A−B or A∖B is the set of all x such that x belongs to A and x does not belong to B.) Set Difference.. • In the previous example: • A = { 1, 2 } • B = { 2, 3 } Set Difference.. • Similarly we can find B−A or B∖A : • (The B−A or B∖A is the set of all x such that x belongs to B and x does not belong to A.) Set Difference.. • In the previous example: • A = { 1, 2 } • B = { 2, 3 } Complement of a Set (Ā) • Sometimes we want to talk about elements which lie OUTSIDE of a given set and within another set. • The set of all those elements which are not contained in a given set is called complement set. • The complement of set A is the set of all elements of the universe which are not in A. Complement of a Set (Ā).. • Symbolically it is represented as Ā or Ã or NOT A. • (Ā is the set of all x such that x does not belong to A.) • It can also be represented as : Complement of a Set.. • In the previous example: • A = { 1, 2 } Complement of a Set ( ) • Similarly we can find the complement of set B which is the set of all elements of the universe which are not in B. • (NOT B ( ) is the set of all x such that x does not belong to B.) • It can also be represented as : Complement of a Set.. • In the previous example: • B = { 2, 3 } Complement of a Union Set A∪B • Similarly we can find the complement of set A∪B which is the set of all elements of the universe which are not in A∪B. • ( is the set of all x such that x does not belong to A∪B.) • It can also be represented as: Complement of a Set A∪B.. • In the previous example: • A = { 1, 2 } • B = { 2, 3 } Complement of a Set A∩B • Similarly we can find the complement of set A∩B which is the set of all elements of the universe which are not in A∩B. • ( is the set of all x such that x does not belong to A∩B.) • It can also be represented as: Complement of a Set A∩B.. • In the previous example: • A = { 1, 2 } • B = { 2, 3 } Disjoint Set • Two sets are said to be disjoint if they have no element in common. • It means their members do not overlap or their intersection is empty set. • If the two sets A and B are disjoint sets then • Example: Let us consider a new example to represent all sets– U, A, and B: • Set U = { 1, 2, 3, 4, 5 } • Set A = { 1, 2 } • Set B = { 3, 4 } Disjoint Set.. • Now the two sets A and B are the disjoint sets because no element is common between these two sets and they are represented as: • Another example is: Subset (B⊂A or B⊆A) • Set B is a subset of set A if and only if every element of set B is also the element of set A. •

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