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The document discusses key concepts about sets, including: 1) Intervals are subsets of real numbers that can be open, closed, or half-open/half-closed. Intervals are represented visually on a number line. 2) The power set of a set A contains all possible subsets of A. Its size is 2 to the power of the size of A. 3) The union of sets A and B contains all elements that are in A, B, or both. The intersection contains all elements common to both sets. 4) Practical problems can use formulas involving the sizes of unions and intersections of finite sets.

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SETS - Vedantu.pdf

The document defines and describes sets. Some key points include:
- A set is a collection of well-defined objects. Sets can be described in roster form or set-builder form.
- There are different types of sets such as the empty/null set, singleton sets, finite sets, and infinite sets.
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The document provides information about sets, relations, and functions. It defines what a set is and how sets can be described in roster form or set-builder form. It discusses different types of sets such as the empty set, singleton sets, finite sets, and infinite sets. It also covers topics like subsets, power sets, operations on sets such as union, intersection, difference, complement, and applications of these concepts to solve problems involving cardinality of sets.

Mathematics JEE quick revision notes pdf

The document provides information about sets, relations, and functions in mathematics:
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- A relation from a set A to a set B is a subset of the Cartesian product A × B. The domain is the set of first elements in the relation and the range is the set of second elements.
- A function from a set A to a set B is a special type of relation where each element of A is mapped to exactly one element of B.

Sets

The document provides information about sets including definitions of key terms like union, intersection, complement, difference, properties of these operations, and counting theorems. It discusses describing sets by explicitly listing members or through a relationship. Examples are provided to illustrate concepts like subsets, proper subsets, power sets, De Morgan's laws, and using Venn diagrams to solve problems involving sets. Counting theorems are presented to calculate the number of elements in unions, intersections, and complements of finite sets.

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This document defines sets and basic set operations. It begins by defining what a set is and providing examples of sets. It then discusses subsets, set equality, Venn diagrams, union, intersection, difference, and complement operations on sets. It provides examples and problems to illustrate these concepts. It also covers empty sets, disjoint sets, partitions of sets, power sets, ordered tuples, Cartesian products, set identities, and proofs using an element argument approach.

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This document provides an overview of set theory concepts including:
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- Finite and countable sets versus infinite sets.
- Product sets involving ordered pairs from two sets.
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The document defines basic concepts about sets including:
- A set is a collection of distinct objects called elements. Sets can be represented using curly brackets or the set builder method.
- Common set symbols are defined such as belongs to (∈), is a subset of (⊆), and is not a subset of (⊄).
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- Laws for sets like commutative, associative, distributive, double complement, and De Morgan's laws are listed.
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The document defines and describes sets. Some key points include:
- A set is a collection of well-defined objects. Sets can be described in roster form or set-builder form.
- There are different types of sets such as the empty/null set, singleton sets, finite sets, and infinite sets.
- Set operations include union, intersection, difference, symmetric difference, and complement. Properties like subsets and the power set are also discussed.
- Cardinality refers to the number of elements in a set. Formulas are given for finding the cardinality of sets under different operations.

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The document provides information about sets, relations, and functions. It defines what a set is and how sets can be described in roster form or set-builder form. It discusses different types of sets such as the empty set, singleton sets, finite sets, and infinite sets. It also covers topics like subsets, power sets, operations on sets such as union, intersection, difference, complement, and applications of these concepts to solve problems involving cardinality of sets.

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The document provides information about sets, relations, and functions in mathematics:
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- A relation from a set A to a set B is a subset of the Cartesian product A × B. The domain is the set of first elements in the relation and the range is the set of second elements.
- A function from a set A to a set B is a special type of relation where each element of A is mapped to exactly one element of B.

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The document provides information about sets including definitions of key terms like union, intersection, complement, difference, properties of these operations, and counting theorems. It discusses describing sets by explicitly listing members or through a relationship. Examples are provided to illustrate concepts like subsets, proper subsets, power sets, De Morgan's laws, and using Venn diagrams to solve problems involving sets. Counting theorems are presented to calculate the number of elements in unions, intersections, and complements of finite sets.

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This document defines sets and basic set operations. It begins by defining what a set is and providing examples of sets. It then discusses subsets, set equality, Venn diagrams, union, intersection, difference, and complement operations on sets. It provides examples and problems to illustrate these concepts. It also covers empty sets, disjoint sets, partitions of sets, power sets, ordered tuples, Cartesian products, set identities, and proofs using an element argument approach.

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- Two sets are equal if they have exactly the same elements. A set A is a subset of set B if every element of A is also an element of B.
- The power set of a set A is the collection of all subsets of A, including the empty set and A itself. It is denoted by P(A).

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This document provides an overview of set theory concepts including:
- Sets, elements, and set operations like union, intersection, difference, and complement.
- Finite and countable sets versus infinite sets.
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- Classes of sets including the power set of a set, which contains all subsets.

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The document defines basic concepts about sets including:
- A set is a collection of distinct objects called elements. Sets can be represented using curly brackets or the set builder method.
- Common set symbols are defined such as belongs to (∈), is a subset of (⊆), and is not a subset of (⊄).
- Types of sets like empty sets, singleton sets, finite sets, and infinite sets are described.
- Operations between sets such as union, intersection, difference, and complement are explained using Venn diagrams.
- Laws for sets like commutative, associative, distributive, double complement, and De Morgan's laws are listed.
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- Common notations are presented for defining sets, elements, membership, subsets, unions, intersections, complements, and cardinality.
- Finite and infinite sets are discussed. Special sets like the empty set, power set, and universal set are introduced.
- Venn diagrams are used to visually represent relationships between sets such as subsets, unions, intersections, and complements.

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#inshallah.

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2. Important sets in number systems include real numbers (IR), positive/negative reals (IR+/IR-), integers (Z), positive/negative integers (Z+/Z-), rational numbers (Q), and natural numbers (N).
3. Sets can be specified using a roster method that lists elements or a set-builder notation that describes elements. Operations on sets include union, intersection, complement, and symmetric difference.

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The document defines sets and provides examples of different types of sets. It discusses how sets can be defined using roster notation by listing elements within braces or using set-builder notation stating properties elements must have. Some key types of sets mentioned include the empty set, natural numbers, integers, rational numbers, and real numbers. Operations on sets like union, intersection, and difference are introduced along with examples. Subsets, Venn diagrams, and the power set are also covered.

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- Algebraic laws governing set operations like distributive, associative, commutative, identity, and De Morgan's laws
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- A set is a well-defined collection of objects or elements. Sets can be defined by listing elements or describing membership rules.
- Common notations are presented for defining sets, elements, membership, subsets, unions, intersections, complements, and cardinality.
- Finite and infinite sets are discussed. Special sets like the empty set, power set, and universal set are introduced.
- Venn diagrams are used to visually represent relationships between sets such as subsets, unions, intersections, and complements.

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This document defines sets and common set operations such as union, intersection, difference, complement, Cartesian product, and cardinality. It begins by defining a set as a collection of distinct objects and provides examples of sets. It then discusses ways to represent and visualize sets using listings, set-builder notation, Venn diagrams, and properties of subsets, supersets, equal sets, disjoint sets, and infinite sets. The document concludes by defining common set operations and identities using membership tables and examples.

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The document provides an overview of sets and logic. It defines basic set concepts like elements, subsets, unions and intersections. It explains Venn diagrams can be used to represent relationships between sets. Logic is introduced as the study of correct reasoning. Propositions are defined as statements that can be determined as true or false. Logical connectives like conjunction, disjunction and negation are explained through truth tables. Compound statements can be formed using these connectives.

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A Brief introduction to set and sets types. in very simple method that can every one enjoy during study.
#inshallah.

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- Algebraic laws governing set operations like distributive, associative, commutative, identity, and De Morgan's laws
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The document provides information about sets including:
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The document defines basic set theory concepts including sets, subsets, unions, intersections, complements, partitions, and Cartesian products. It provides examples to demonstrate set equality, proving and disproving subset relations, finding unions and intersections of sets, and applying set operations and concepts to numbers, intervals, and Cartesian products of sets.

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- 1. Class XI - MATHEMATICS Chapter 1 – SETS Module – 2/2 By Smt. Mini Maria Tomy PGT Mathematics AECS KAIGA Distance Learning Programme: An initiative by AEES, Mumbai
- 2. Learning Outcome In this module we are going to learn about • Intervals as subset of R • Power set • Universal set • Venn diagrams • Union of sets • Intersection of sets • Practical problems on Union & Intersection of sets
- 3. INTERVALS AS SUBSET OF REAL NUMBERS Open Interval: Let a, b ∈ R and a < b. Then the set of real numbers { x : a < x < b} is called an open interval and is denoted by (a, b). Closed Interval: Let a, b ∈ R and a < b. Then the set of real numbers {x : a ≤ x ≤ b} is called closed interval and is denoted by [ a, b ].
- 4. TYPES OF INTERVALS (a, b) = { x : a < x < b, x ∈ R} [a, b] = {x : a ≤ x ≤ b, x ∈ R} (a, b] = { x : a < x ≤ b, x ∈ R} [a, b) = { x : a ≤ x < b, x ∈ R} (0, ∞) = { x : 0 < x < ∞, x ∈ R} (- ∞,∞) = {x: -∞ < x < ∞, x ∈ R} = the set of real numbers R.
- 5. REPRESENTING INTERVALS ON NUMBER LINE (a, b) [a, b] (a, b] [a, b) a b a b a b a b ⇒ ⇒ ⇒ ⇒
- 6. POWER SET The collection of all subsets of a set A is called the power set of A. It is denoted by P(A). if A = { 1, 2 }, then P( A ) = { φ,{ 1 }, { 2 }, { 1,2 }} In general, if A is a set with n(A) = m, then n [ P(A)] = 2m.
- 7. UNIVERSAL SET In a particular context, we have to deal with the elements and subsets of a basic set which is relevant to that particular context. This basic set is called the “Universal Set”. The universal set is usually denoted by U, and all its subsets by the letters A, B, C, etc. For example, for the set of all integers, the universal set can be the set of rational numbers or the set R of real numbers.
- 8. VENN DIAGRAMS Most of the relationships between sets can be represented by means of diagrams which are known as Venn diagrams. Venn diagrams are named after the English logician, John Venn (1834-1883). In a Venn diagram, the universal set is represented usually by a rectangle and its subsets by circles or ellipses. John Venn (1834-1883).
- 9. UNION OF SETS The union of two sets A and B is the set C which consists of all those elements which are either in A or in B (including those which are in both). A B U A ∪ B The union of two sets A and B is denoted as A ∪ B and usually read as ‘A union B’. Hence, A ∪ B = { x : x ∈A or x ∈B }
- 10. Some examples on union of two sets Example 1 : Let A = { 2, 4, 6, 8} and B = { 6, 8, 10, 12}. Find A ∪ B Solution: A ∪ B = { 2, 4, 6, 8, 10, 12} Example 2 : Let A = { a, b, c, d } and B = { a, c}. Find A ∪ B Solution: A ∪ B = {a, b, c, d } Note : If, B ⊂ A, then A ∪ B = A.
- 11. PROPERTIES OF UNION OF TWO SETS i) A ∪ B = B ∪ A (Commutative law) ii) (A ∪ B) ∪ C = A ∪ (B ∪ C) (Associative law ) iii) A ∪ ∅ = A (Law of identity element, ∅ is the identity of ∪) iv) A ∪ A = A (Idempotent law) v) U ∪ A = U (Law of U)
- 12. INTERSECTION OF TWO SETS The intersection of two sets A and B is the set of all those elements which belong to both A and B. Symbolically, we write A ∩ B = {x : x ∈ A and x ∈ B}. Example: Let A = {1, 3, 4, 5, 6, 7, 8} and B = { 2, 3, 5, 7, 9}, find A∩B. Solution: A∩B = {3, 5, 7}. Note: If A ⊂ B ,then A∩B = B A ∩ B U A B
- 13. PROPERTIES OF INTERSECTION OF TWO SETS i) A ∩ B = B ∩ A (Commutative law). ii) ( A ∩ B ) ∩ C = A ∩ ( B ∩ C ) (Associative law). iii) ∅ ∩ A = ∅ , U ∩ A = A (Law of ∅ and U). iv) A ∩ A = A (Idempotent law) v) A∩( B ∪ C) = ( A ∩ B ) ∪ ( A ∩ C ) (Distributive law )
- 14. Distribution of Intersection over Union B ∪ C A∩ (B∪ C ) A ∩ B A ∩ C ( A ∩ B ) ∪ ( A ∩ C )
- 15. Practical problems on Union & Intersection of sets if A and B are finite sets, then n ( A ∪ B ) = n ( A ) + n ( B ) – n ( A ∩ B ) If A ∩ B = φ, then, n ( A ∪ B ) = n ( A ) + n ( B ) If A, B and C are finite sets, then n(A∪ B∪ C) = n(A) + n(B) + n(C) – n (A∩B ) – n(B∩C) – n(A∩C) + n(A∩B∩C)
- 16. Example In a school there are 20 teachers who teach mathematics or physics. Of these, 12 teach mathematics and 4 teach both physics and mathematics. How many teach physics ? Solution:Let M denote the set of teachers who teach mathematics and P denote the set of teachers who teach physics. Then, n(M∪P ) = 20, n(M ) = 12 , n(M∩P ) = 4, n(P)= ? n ( M ∪ P ) = n ( M ) + n ( P ) – n ( M ∩ P ), we obtain 20 = 12 + n ( P ) – 4 Thus n ( P ) = 12 Hence 12 teachers teach physics.
- 17. What have we learned today? Intervals are subsets of R. The power set P(A) of a set A is the collection of all subsets of A. The union of two sets A and B is the set of all those elements which are either in A or in B. The intersection of two sets A and B is the set of all elements which are common in A and B. If A and B are finite sets then n(A∪B) = n(A)+n(B) – n(A∩B) If A ∩ B = φ, then n (A ∪ B) = n (A) + n (B) .
- 18. THANK YOU!