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Class 11 Mathematics · Chapter 1
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Physics
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Physics
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History
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Geography
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Geography
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Geography
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Political Science
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सृजनात्मक लेखन और अनुवाद
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हिंदुस्तानी संगीत (गायन / वादन)
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हिंदुस्तानी संगीत (गायन / वादन)
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हिंदुस्तानी संगीत (तबला / पखावज)
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Summary
A set is a well-defined collection of distinct objects, so one can always decide whether a given object belongs to it. Sets are written in roster form, listing elements within braces, or in set-builder form, stating a common property such as \(\{x : x\text{ is a vowel}\}\). Special number sets \(\mathbb{N},\mathbb{Z},\mathbb{Q},\mathbb{R}\) recur throughout the book. The empty set \(\varphi\) has no element; a set is finite or infinite by its count \(n(S)\). Two sets are equal when they have exactly the same elements, and \(A\subset B\) means every element of \(A\) lies in \(B\). Intervals like \((a,b)\) and \([a,b]\) are subsets of \(\mathbb{R}\). Operations on sets — union \(A\cup B\), intersection \(A\cap B\), difference \(A-B\) and complement \(A'\) with respect to a universal set \(U\) — obey commutative, associative, distributive, idempotent and De Morgan's laws \((A\cup B)'=A'\cap B'\). Venn diagrams picture these relationships, making the theory of sets the foundation for relations, functions and probability that follow.
Sets and their representationsEmpty, finite and infinite setsEqual sets and subsetsIntervals as subsets of \(\mathbb{R}\)Universal set and Venn diagramsUnion, intersection, difference and complementDe Morgan's laws
Key terms
Set
A well-defined collection of distinct objects; membership can always be decided.
Empty set
A set with no elements, denoted \(\varphi\) or \(\{\}\).
Subset
\(A\subset B\) when every element of \(A\) is in \(B\).
Universal set
The fixed set \(U\) containing all elements relevant to the discussion.
Union and intersection
\(A\cup B=\{x:x\in A\text{ or }x\in B\}\); \(A\cap B=\{x:x\in A\text{ and }x\in B\}\).
Complement
\(A'=U-A=\{x:x\in U,\ x\notin A\}\).
Extra questions & answers
Try one from this chapter
Let A = {1, 2, 3, 4} and B = {3, 4, 5, 6}. What is A ∩ B?