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Title: Set Theory - Class 11
Introduction:
- Set theory is a branch of mathematical logic that deals with the study of sets, which are collections of distinct objects.
- Sets are fundamental to various areas of mathematics and provide a foundation for many other mathematical concepts.
Basic Concepts:
1. Sets: A set is a well-defined collection of distinct objects, called elements or members of the set.
- Notation: Sets are denoted by capital letters (e.g., A, B, C) and elements by lowercase letters (e.g., a, b, c).
- Example: A = {1, 2, 3} represents a set A with elements 1, 2, and 3.
2. Representation of Sets:
- Roster Method: In this method, elements of a set are listed within curly braces.
- Set-Builder Form: In this method, sets are defined using a property or condition that the elements must satisfy.
- Example: B = {x | x is an even number} represents a set B consisting of all even numbers.
3. Cardinality: The cardinality of a set is the number of elements it contains.
- Notation: The cardinality of set A is represented by |A|.
- Example: If A = {1, 2, 3}, then |A| = 3.
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Set Operations:
1. Union: The union of two sets A and B, denoted by A ∪ B, is a set that contains all elements present in either A or B.
- Example: If A = {1, 2, 3} and B = {2, 3, 4}, then A ∪ B = {1, 2, 3, 4}.
2. Intersection: The intersection of two sets A and B, denoted by A ∩ B, is a set that contains common elements of A and B.
- Example: If A = {1, 2, 3} and B = {2, 3, 4}, then A ∩ B = {2, 3}.
3. Difference: The difference between two sets A and B, denoted by A - B, is a set that contains elements of A not present in B.
- Example: If A = {1, 2, 3} and B = {2, 3, 4}, then A - B = {1}.
4. Complement: The complement of a set A, denoted by A', is a set that contains all elements not present in A.
- Example: If U is the universal set and A = {1, 2, 3}, then A' = U - A.
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Set Relations:
1. Equal Sets: Two sets A and B are said to be equal if they have the same elements.
- Example: If A = {1, 2, 3} and B = {3, 1, 2}, then A = B.
2. Subsets: A set A is said to be a subset of another set B if every element of A is also an element of B.
- Notation: A ⊆ B (A is a subset of B)
- Example: If A = {1, 2} and B = {1, 2, 3}, then A ⊆ B.
3. Power Set: The power set of a set A is the set of all possible subsets of A.
- Notation: P(A)
- Example: If A = {1, 2}, then P(A)
= {{}, {1}, {2}, {1, 2}}.
4. Cartesian Product: The Cartesian product of two sets A and B, denoted by A × B, is a set of ordered pairs (a, b) where a belongs to A and b belongs to B.
- Example: If A = {1, 2} and B = {3, 4}, then A × B = {(1, 3), (1, 4), (2, 3), (2, 4)}.
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Questions (Types):
1. Multiple Choice Questions:
a) Which of the following is a set in the roster method notation?
i) A = {x | x is a prime number}
ii) B = {1, 1, 2, 3, 5}
iii) C = {x | x is a multiple of 4}
b) If A = {1, 2, 3} and B = {2, 3, 4}, find A ∪ B.
2. True/False Questions:
a) A ⊆ B if and only if A = B.
b) A ∩ B = B ∩ A for any two sets A and B.
3. Fill in the Blanks:
a) The number of elements in a power set with n elements is ______.
b) The intersection of two sets A and B is denoted by ______.
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4. Short Answer Questions:
a) Define the complement of a set and provide an example.
b) What is the cardinality of the set of all natural numbers?
5. Problem-Solving Questions:
a) If A = {1, 2, 3, 4} and B = {2, 3, 4, 5}, find A - B and B - A.
b) Let A = {1, 2, 3, 4, 5} and B = {3, 4, 5, 6, 7}. Find (A ∪ B)'.
Note: The provided questions cover a range of question types commonly encountered in set theory, including multiple choice, true/false, fill in the blanks, short answer, and problem-solving. These questions will help reinforce the concepts discussed in the chapter and test the understanding of the students.