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- 1. LR DAV PUBLIC SCHOOL, GANDARPUR, CUTTACK. MATHEMATICS CLASS-XI SETS BY SUSHREE S. SAHU
- 2. INTRODUCTION: The theory of sets was developed by German mathematicians George Cantor. Sets forms the basis of several other fields of study like Relations and functions, geometry, sequences, probability etc.
- 3. DEFINITION: A set is a well-defined collection of objects. The objects are called as elements or members of the set. Example: (i) The rivers of India. (ii) The vowels of English alphabet. (iii) The factors of 12, namely 1, 2, 3, 4, 6, 12. The above examples are well defined collection of objects in a particular category. NOTE: “The set of good cricketers in the World” is not a set because the word “good” is not well defined here. The criteria to define goodness varies from person to person.
- 4. REPRESENTATION OF A SET: Sets are usually represented by capital letters like A, B, C, D, M, N etc. The elements of a set are represented by small letters a, b, c, x, y, z etc. The elements are separated by commas and are enclosed within the curly braces{}. Example: M={a, b, c, d} Here M is a set which contains the elements a, b, c, d. If ‘a’ is an element of set M then we say that ‘a’ belongs to the set M. We write it as a∈ M. If ‘p’ is not an element of set M then we say that ‘p’ does not belong to the set M. We write it as p∉ M.
- 5. METHODS OF REPRESENTATION OF A SET: There are two methods of representing a set: (i) Roaster or tabular form (ii) Set builder form (i) Roaster or tabular form: In Roaster form, all the elements of a set are enlisted being separated by commas and enclosed within curly braces. Ex- If A is the set of all prime numbers less than 10, then A={2, 3, 5, 7} NOTE: The order in which the elements are written in a set makes no difference. Also the repetition of an element has no effect. Ex-{1, 2, 3} is same as {3, 2, 1}. Also {1,2,3,1,3} is same as {1,2,3}.
- 6. (ii) Set-builder form: In Set-builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set. Ex-The set of all even natural number less than 10 can be written as: A={𝑥: 𝑥 is an even natural number less than 10} Or A={𝑥: 𝑥 = 2𝑛, 𝑛 ∈ 𝑁 and 𝑛 < 5} The set A can be read as “the set of all 𝑥 such that 𝑥 is an even natural number less than 10”. This set can be represented in roaster form i.e. A={2,4,6,8}
- 7. Examples: Converting set builder form to roaster form: (i) A={𝑥: 𝑥 is an odd natural number} A={1, 3, 5, 7,……..} (ii) B={𝑥: 𝑥 is a letter in the word “MATHEMATICS”} B={M, A, T, H, E, I, C,S} Converting roaster form to set builder form: (i)A={1, 4, 9, 16, ……..} A={𝑥: 𝑥 = 𝑛2,where 𝑛 ∈ 𝑁} (ii) B={5, 25,125, 625} B={𝑥: 𝑥 = 5𝑛,1 ≤ 𝑛 ≤ 4}
- 8. Some important number sets:- 𝑁 = The s𝑒𝑡 𝑜𝑓 𝑎𝑙𝑙 𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 = 1, 2, 3, 4, 5, … … … … … 𝑊 = 𝑇ℎ𝑒 𝑠𝑒𝑡 𝑜𝑓 𝑎𝑙𝑙 𝑤ℎ𝑜𝑙𝑒 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 = 0, 1, 2, 3, 4, … … … … … . 𝑍 = 𝑇ℎ𝑒 𝑠𝑒𝑡 𝑜𝑓 𝑎𝑙𝑙 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠 = … … … … . . −3, −2, −1, 0, 1, 2, 3, … … … … 𝑄 = 𝑇ℎ𝑒 𝑠𝑒𝑡 𝑜𝑓 𝑎𝑙𝑙 𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 = 𝑝 𝑞 : 𝑝, 𝑞 ∈ 𝑍, 𝑞 ≠ 0 = 1, 2, 1 3 , 5 3 , 0, − 7 9 , … … … … . . 𝑅 = 𝑇ℎ𝑒 𝑠𝑒𝑡 𝑜𝑓 𝑎𝑙𝑙 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑄′ = 𝑅 − 𝑄 = 𝑇 = 𝑥: 𝑥 ∈ 𝑅 𝑎𝑛𝑑 𝑥 ∉ 𝑄 = 𝑇ℎ𝑒 𝑠𝑒𝑡 𝑜𝑓 𝑎𝑙𝑙 𝑖𝑟𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠, 𝑖. 𝑒 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑤ℎ𝑖𝑐ℎ 𝑎𝑟𝑒 𝑛𝑜𝑡 𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 = 2, 3, 2 + 5, … … … … . 𝜋, 𝑒, log 2 … … … … … .
- 9. DIFFERENT TYPES OF SETS: The Empty Set: A set containing no elements is called an Empty set or Null set or void set. It is denoted by ∅(Read as phi) or { }. Example:- 𝐴 = 𝑥 ∈ 𝑁: 1 < 𝑥 < 2 = ∅. Singleton Set: A set consisting of a single element is called a singleton set. Example:- A= 𝑥 ∈ 𝑁: 𝑥2 = 9 is a singleton set as A={3}. Finite Set: A set which is empty or consists of a definite number of elements is called a finite set. Example:- A={1,3,5} is a finite set. Cardinal number of a finite set: The number of elements present in the finite set is called the cardinal number of a that set. It is denoted by n(A). Ex- The cardinal number of the above set A is 3 i.e. n(A)=3
- 10. Infinite Set: If the set does not contain finite number of elements then the set is called an infinite sets. Ex- The set of all lines in a plane. Equal Sets: Two sets A and B are said to be equal if every elements of set A is a member of set B and every elements of set B is a member of set A. Ex-1. A={1,2,3,4} and B={4,3,2,1} are equal sets. 2. A={x: x is a letter of the word “ITEM”} and B={x: x is a letter of the word “TIME”} are equal sets. Equivalent Sets: Two finite sets A and B are said to be equivalent sets if their cardinal numbers are same i.e. if n(A)=n(B).
- 11. Subsets: Proper Subset: Let A and B be two sets. Set 𝐴 is called a proper subset of set 𝐵 if all the elements of set 𝐴 are also in set 𝐵 and there is at least one more element in 𝐵 which is not in 𝐴. We denote this as 𝑨 ⊂ 𝑩. Ex- A={1,2,3,4}, B={1,2,3,4,5,6}. Here 𝑨 ⊂ 𝑩. Subset: A set 𝐴 is called a subset of another set 𝐵 if 𝐴 is a proper subset of 𝐵 or A is equal to 𝐵. We denote it by 𝑨 ⊆ 𝑩. Also we read this as "𝐴 is subset of 𝐵". Also we can say here "𝐵 is superset of 𝐴". We denote it by 𝑩 ⊇ 𝑨. Important results: The Empty set (∅) is a subset of every set. Every set is subset of itself. If 𝑨 ⊆ 𝑩 and 𝑩 ⊇ 𝑨 then 𝑨 = 𝑩. The total no. of subsets of a set containing ‘𝑛’ number of elements is 2𝑛. The total no. of non-empty subsets of a set containing ‘𝑛’ number of elements is 2𝑛−1. B A 𝑨 ⊂ 𝑩
- 12. INTERVALS AS SUBSETS OF R: Closed Intervals: The set of all numbers 𝑥 such that 𝒂 ≤ 𝒙 ≤ 𝒃 is called a closed interval. It is denoted by [𝒂, 𝒃]. [𝒂, 𝒃]={𝒙: 𝒂 ≤ 𝒙 ≤ 𝒃 } i.e. the set contains all numbers in between 𝒂 and 𝒃 including 𝒂 and 𝒃 itself. Open Intervals: The set of all numbers 𝑥 such that 𝒂 < 𝒙 < 𝒃 is called a closed interval. It is denoted by (𝒂, 𝒃). (𝒂, 𝒃)={𝒙: 𝒂 < 𝒙 < 𝒃 } i.e. the set contains all numbers in between 𝒂 and 𝒃 excluding 𝒂 and 𝒃. a b [a, b] a b (a, b)
- 13. Semi Open/Closed Intervals(opened at left end and closed at right end): The set of all numbers 𝑥 such that 𝒂 < 𝒙 ≤ 𝒃 is called a semi open/closed interval. It is denoted by (𝒂, 𝒃]. (𝒂, 𝒃]={𝒙: 𝒂 < 𝒙 ≤ 𝒃 } i.e. the set contains all numbers in between 𝒂 and 𝒃 excluding 𝒂 but including 𝒃. Semi Open/Closed Intervals( closed at left end and opened at right end): The set of all numbers 𝑥 such that 𝒂 ≤ 𝒙 < 𝒃 is called a semi open/closed interval. It is denoted by [𝒂, 𝒃). [𝒂, 𝒃)={𝒙: 𝒂 ≤ 𝒙 < 𝒃 } i.e. the set contains all numbers in between 𝒂 and 𝒃 including 𝒂 but excluding 𝒃. a b (a, b] a b [a, b)
- 14. UNIVERSAL set: A set that contains all sets in a given context is called the universal set and is denoted by U. Ex-When we are using intervals on real line , the set R of real numbers is taken as the universal set. POWER set: Let A be a set. Then the set containing all the subsets of set A is called the power set of A and is denoted by P(A). P(A)={S: S⊂A}. Ex- Let A={1,2}.Then P(A)={{1},{2},{1,2}, ∅}. Note: 1.Since the empty set and the set A itself are subsets of A and therefore elements of P(A).Thus the power set of a set is always non-empty. 2.If A is a finite set having 𝑛 elements, then P(A) has 2𝑛 elements.
- 15. VENN DIAGRAM First of all a Swiss mathematician Euler represented the set by diagrams. Then British mathematician John Venn popularized the diagram in the 1880s. Thus the diagrammatic or pictorial representation of sets are called as Venn-Euler Diagram or simply Venn Diagram. In Venn-Diagram, the Universal set Uis represented by points within a rectangle and its subsets are represented by points in closed curves(usually circles) within the rectangle. If set A is subset of a set B, then the circle representing the set A is drawn inside the circle representing set B. U B A
- 16. If A and B are not equal but they have some common elements, then to represent A and B we draw two intersecting circles inside the rectangle representing the universal set. Two disjoint sets are represented by two non-intersecting circles. U B A U A B
- 17. OPERATIONS ON SETS: UNION: Let A and B be two sets. The union of A and B is the set of all those elements that either belongs to A or belongs to B or to both A and B. A∪B={𝑥: 𝑥 ∈ 𝐴 𝑜𝑟 𝑥 ∈ 𝐵} We can say if 𝑥 ∈ A ∪ B ⇔ 𝑥 ∈ A 𝑜𝑟 𝑥 ∈ B. Ex-If A={a, b, c, d} and B={d, e, f}, then A∪B={a, b, c, d, e, f}. Properties: 𝐴 ⊂ 𝐴 ∪ 𝐵 𝑎𝑛𝑑 𝐵 ⊂ 𝐴 ∪ 𝐵 . If 𝐴 ⊂ 𝐵,then 𝐴 ∪ 𝐵 = 𝐵 and if 𝐵 ⊂ 𝐴,then 𝐴 ∪ 𝐵 = 𝐴. 𝐴 ∪ 𝐵 = 𝐵 ∪ 𝐴(Commutative law) 𝐴 ∪ 𝐵 ∪ 𝐶 = 𝐴 ∪ (𝐵 ∪ 𝐶)(Associative law) 𝐴 ∪ ∅ = 𝐴(Identity law) 𝐴 ∪ 𝐴 = 𝐴(Idempotent law) 𝐴 ∪ 𝑈 = 𝑈(Law of U) A B B A A∪B, if A and B are joint sets. A∪B, if A and B are disjoint sets. A∪B, if A ⊂B
- 18. INTERSECTION: Let A and B be two sets. The intersection of A and B is the set of all those elements that belongs to both A and B. A∩B={𝑥: 𝑥 ∈ 𝐴 𝑎𝑛𝑑 𝑥 ∈ 𝐵} We can say if 𝑥 ∈ A ∩ B ⇔ 𝑥 ∈ A 𝑎𝑛𝑑 𝑥 ∈ B. Ex-If A={a, b, c, d} and B={d, e, f}, then A∩B={ d }. Properties: 𝐴 ∩ 𝐵 ⊂ 𝐴 𝑎𝑛𝑑 𝐴 ∩ 𝐵 ⊂ 𝐵. If 𝐴 ⊂ 𝐵,then 𝐴 ∩ 𝐵 = 𝐴 and if 𝐵 ⊂ 𝐴,then 𝐴 ∩ 𝐵 = 𝐵. 𝐴 ∩ 𝐵 = 𝐵 ∩ 𝐴(Commutative law) 𝐴 ∩ 𝐵 ∩ 𝐶 = 𝐴 ∩ (𝐵 ∩ 𝐶)(Associative law) 𝐴 ∩ ∅ = ∅(Identity law) 𝐴 ∩ 𝐴 = 𝐴(Idempotent law) 𝐴 ∩ 𝑈 = 𝐴(Law of U) A∩B, if A and B are joint sets. A B A∩B=∅, if A and B are disjoint sets. B A A∩B=A if A ⊂ B .
- 19. • DISTRIBUTIVE LAW: 1.Union is distributive over intersection i.e. for set 𝐴, 𝐵, 𝐶 ; 𝐴 ∪ 𝐵 ∩ 𝐶 = 𝐴 ∪ 𝐵 ∩ 𝐴 ∪ 𝐶 . Proof: Let 𝑥 ∈ 𝐴 ∪ 𝐵 ∩ 𝐶 ⇔ 𝑥 ∈ 𝐴 𝑜𝑟 𝑥 ∈ 𝐵 ∩ 𝐶 ⇔ 𝑥 ∈ 𝐴 𝑜𝑟 𝑥 ∈ 𝐵 𝑎𝑛𝑑 𝑥 ∈ 𝐶 ⇔ 𝑥 ∈ 𝐴 𝑜𝑟 𝑥 ∈ 𝐵 𝑎𝑛𝑑 𝑥 ∈ 𝐴 𝑜𝑟 𝑥 ∈ 𝐶 ⇔ 𝑥 ∈ 𝐴 ∪ 𝐵 𝑎𝑛𝑑 𝑥 ∈ 𝐴 ∪ 𝐶 ⇔ 𝑥 ∈ 𝐴 ∪ 𝐵 ∩ 𝐴 ∪ 𝐶 so 𝐴 ∪ 𝐵 ∩ 𝐶 ⊆ 𝐴 ∪ 𝐵 ∩ 𝐴 ∪ 𝐶 and 𝐴 ∪ 𝐵 ∩ 𝐴 ∪ 𝐶 ⊆ 𝐴 ∪ 𝐵 ∩ 𝐶 . Hence 𝐴 ∪ 𝐵 ∩ 𝐶 = 𝐴 ∪ 𝐵 ∩ 𝐴 ∪ 𝐶 .
- 20. 2.Intersection is distributive over union i.e. for set 𝐴, 𝐵, 𝐶 ; 𝐴 ∩ 𝐵 ∪ 𝐶 = 𝐴 ∩ 𝐵 ∪ 𝐴 ∩ 𝐶 Proof: Let 𝑥 ∈ 𝐴 ∩ 𝐵 ∪ 𝐶 ⇔ 𝑥 ∈ 𝐴 𝑎𝑛𝑑 𝑥 ∈ 𝐵 ∪ 𝐶 ⇔ 𝑥 ∈ 𝐴 𝑎𝑛𝑑 𝑥 ∈ 𝐵 𝑜𝑟 𝑥 ∈ 𝐶 ⇔ 𝑥 ∈ 𝐴 𝑎𝑛𝑑 𝑥 ∈ 𝐵 𝑜𝑟 𝑥 ∈ 𝐴 𝑎𝑛𝑑 𝑥 ∈ 𝐶 ⇔ 𝑥 ∈ 𝐴 ∩ 𝐵 𝑜𝑟 𝑥 ∈ 𝐴 ∩ 𝐶 ⇔ 𝑥 ∈ 𝐴 ∩ 𝐵 ∪ 𝐴 ∩ 𝐶 So 𝐴 ∩ 𝐵 ∪ 𝐶 ⊆ 𝐴 ∩ 𝐵 ∪ 𝐴 ∩ 𝐶 and 𝐴 ∩ 𝐵 ∪ 𝐴 ∩ 𝐶 ⊆ 𝐴 ∩ 𝐵 ∪ 𝐶 . Hence 𝐴 ∩ 𝐵 ∪ 𝐶 = 𝐴 ∩ 𝐵 ∪ 𝐴 ∩ 𝐶 .
- 21. Difference of sets: Let A and B be two sets. The difference of A and B, written as A−B contains the elements which belong to set A but do not belong to set B. A−B={𝑥: 𝑥 ∈ 𝐴 𝑎𝑛𝑑 𝑥 ∉ 𝐵} We can say if 𝑥 ∈ A − B ⇔ 𝑥 ∈ A 𝑎𝑛𝑑 𝑥 ∉ B. Similarly the set B-A contains the elements that are present only in set B but not in set A. B−A={𝑥: 𝑥 ∈ 𝐵 𝑎𝑛𝑑 𝑥 ∉ 𝐴} We can say if 𝑥 ∈ B − A ⇔ 𝑥 ∈ B 𝑎𝑛𝑑 𝑥 ∉ A. Ex-If A={a, b, c, d} and B={d, e, f}, then A−B={a, b, c} and B−A={e, f}.
- 22. Symmetric Difference of two sets: Let A and B be two sets. The symmetric difference of sets A and B is the set (A−B)∪(B−A) and is denoted by A∆B. A∆B= (A−B)∪(B−A)={𝑥: 𝑥 ∉ A ∩ B} Also we can write it as: A∆B=(A∪B)−(A∩B). The shaded region is A∆B
- 23. COMPLEMENT of A set: Let U be the universal set and let A be a set such that A⊂U. Then the complement of A with respect to U is the set that contains all the elements which are not in the set A. It is denoted by A′ or A𝑐 or U-A. A′={𝑥: 𝑥 ∉ 𝐴} We can say that 𝑥 ∈ A′ ⇔ 𝑥 ∉ A. Properties: 𝐴 ∪ 𝐴′ = 𝑈 𝐴 ∩ 𝐴′ = ∅ 𝑈′ = ∅ ∅′ = 𝑈 𝐴′ ′ = 𝐴 De-Morgan’s Law: If A and B be two sets then, (i) 𝐴 ∪ 𝐵 ′ = 𝐴′ ∩ 𝐵′ (ii) 𝐴 ∩ 𝐵 ′ = 𝐴′ ∪ 𝐵′ . A A′ U
- 24. Proof of De-Morgan’s Law: (i) 𝐴 ∪ 𝐵 ′ = 𝐴′ ∩ 𝐵′ Proof: Let 𝑥 ∈ 𝐴 ∪ 𝐵 ′ ⇔ 𝑥 ∉ 𝐴 ∪ 𝐵 ⇔ 𝑥 ∉ 𝐴 𝑎𝑛𝑑 𝑥 ∉ 𝐵 ⇔ 𝑥 ∈ 𝐴′ 𝑎𝑛𝑑 𝑥 ∈ 𝐵′ ⇔ 𝑥 ∈ 𝐴′ ∩ 𝐵′ So 𝐴 ∪ 𝐵 ′ ⊆ 𝐴′ ∩ 𝐵′ and 𝐴′ ∩ 𝐵′ ⊆ 𝐴 ∪ 𝐵 ′ Hence 𝐴 ∪ 𝐵 ′ = 𝐴′ ∩ 𝐵′ (ii) 𝐴 ∩ 𝐵 ′ = 𝐴′ ∪ 𝐵′ Proof: Let 𝑥 ∈ 𝐴 ∩ 𝐵 ′ ⇔ 𝑥 ∉ 𝐴 ∩ 𝐵 ⇔ 𝑥 ∉ 𝐴 𝑜𝑟 𝑥 ∉ 𝐵 ⇔ 𝑥 ∈ 𝐴′ 𝑜𝑟 𝑥 ∈ 𝐵′ ⇔ 𝑥 ∈ 𝐴′ ∪ 𝐵′ So 𝐴 ∩ 𝐵 ′ ⊆ 𝐴′ ∪ 𝐵′ and 𝐴′ ∪ 𝐵′ ⊆ 𝐴 ∩ 𝐵 ′ Hence 𝐴 ∩ 𝐵 ′ = 𝐴′ ∪ 𝐵′
- 25. SOME IMPORTANT RESULTS ON NUMBER OF ELEMENTS IN SETS: Let 𝐴 𝑎𝑛𝑑 𝐵 be finite sets. If 𝐴 ∩ 𝐵 = ∅(i. e The sets 𝐴 𝑎𝑛𝑑 𝐵 are disjoint), then 𝑛 𝐴 ∪ 𝐵 = 𝑛 𝐴 + 𝑛(𝐵) In general, Let 𝐴 𝑎𝑛𝑑 𝐵 be finite sets, then 𝑛 𝐴 ∪ 𝐵 = 𝑛 𝐴 + 𝑛 𝐵 − 𝑛 𝐴 ∩ 𝐵 If 𝐴, 𝐵 𝑎𝑛𝑑 𝐶 are finite sets, then 𝑛 𝐴 ∪ 𝐵 ∪ 𝐶 = 𝑛 𝐴 + 𝑛 𝐵 + 𝑛 𝐶 − 𝑛(𝐴 ∩
- 26. SOME PRACTICAL PROBLEMS: 1.In a committee,50 people speak French,20 people speak Spanish and 10 people speak both Spanish and French. How many speak at least one of these two languages? Ans: Let F denote the set of people who speak French and S denote the set of people who speak Spanish. It is given n(F)=50, n(S)=20 and n(F∩S)=10. To find the number of people who speak at least one of these two languages means to find n(F∪S). We know that, n(F∪S)=n(F)+n(S)-n(F∩S) =50+20-10 =60 Hence 60 people speak at least one of these two languages.
- 27. 2.A college awarded 38 medals in Football, 15 in Basketball and 20 to Cricket. If these medals went to a total of 58 men and only 3 men got medals in all the three sports, how many received medals in exactly two of the three sports? Ans: Let F denote the set of men who received medals in Football, B the set of men who received medals in Basketball and C the set of men who received medals in Cricket. It is given that 𝑛 𝐹 = 38, 𝑛 𝐵 = 15, 𝑛 𝐶 = 20, 𝑛 𝐹 ∪ 𝐵 ∪ 𝐶 = 58 𝑎𝑛𝑑 𝑛 𝐹 ∩ 𝐵 ∩ 𝐶 = 3. Now, 𝑛 𝐹 ∪ 𝐵 ∪ 𝐶 = 𝑛 𝐹 + 𝑛 𝐵 + 𝑛 𝐶 − 𝑛 𝐹 ∩ 𝐵 − 𝑛 𝐵 ∩ 𝐶 − 𝑛 𝐹 ∩ 𝐶 + 𝑛 𝐹 ∩ 𝐵 ∩ 𝐶 ⇒ 58 = 38 + 15 + 20 − 𝑛 𝐹 ∩ 𝐵 − 𝑛 𝐵 ∩ 𝐶 − 𝑛 𝐹 ∩ 𝐶 +3 ⇒ 𝑛 𝐹 ∩ 𝐵 + 𝑛 𝐵 ∩ 𝐶 + 𝑛 𝐹 ∩ 𝐶 = 76 − 58 = 18 Number of men who received medals in exactly two of the three sports; = 𝑛 𝐹 ∩ 𝐵 + 𝑛 𝐵 ∩ 𝐶 + 𝑛 𝐹 ∩ 𝐶 − 3 𝑛 𝐹 ∩ 𝐵 ∩ 𝐶 = 18 − 3 × 3 = 9. Thus, 9 men received medals in exactly two of the three sports.