What went wrong when the king rode under the arch?

It was too low and knocked off his crown

How does the chain of blame unfold in the poem?

Each accused person blames another, ending with the king's own changes

What is a set? Give the two ways of representing it.

A set is a well-defined collection of distinct objects. It is represented in roster (tabular) form by listing elements in braces, or in set-builder form by stating a common property of its elements.

Define the empty set and give one example.

A set with no element is the empty set \(\varphi\). Example: \(\{x:x^2-2=0,\ x\text{ is rational}\}=\varphi\) since \(\sqrt2\) is irrational.

When are two sets said to be equal?

Two sets \(A\) and \(B\) are equal, \(A=B\), if they have exactly the same elements; repetition and order of listing do not matter.

State the number of subsets of a finite set with \(n\) elements.

A set with \(n\) elements has \(2^{n}\) subsets, of which \(2^{n}-1\) are proper subsets.

Write \(A=\{x:x\in\mathbb{R},\ -4<x\le 6\}\) as an interval.

It is the interval \((-4,6]\), open at \(-4\) and closed at \(6\); its length is \(6-(-4)=10\).

State De Morgan's laws.

\((A\cup B)'=A'\cap B'\) and \((A\cap B)'=A'\cup B'\): the complement of a union is the intersection of complements and vice versa.

Distinguish a finite from an infinite set.

A set is finite if it is empty or its elements can be counted to a definite natural number \(n(S)\); otherwise it is infinite, e.g. \(\mathbb{N}\).

Define disjoint sets.

Sets \(A\) and \(B\) are disjoint if \(A\cap B=\varphi\), i.e. they have no element in common, e.g. \(\{2,4,6\}\) and \(\{1,3,5\}\).

List the complement laws for a subset \(A\) of \(U\).

\(A\cup A'=U\), \(A\cap A'=\varphi\), \((A')'=A\), \(\varphi'=U\) and \(U'=\varphi\).

Prove that \(A\cup(A\cap B)=A\) (absorption law).

Every element of \(A\cap B\) already lies in \(A\), so the union adds nothing new; hence \(A\cup(A\cap B)=A\).

Sets

Q: Write \(A=\{x:x\in\mathbb{R},\ -4<x\le 6\}\) as an interval.

It is the interval \((-4,6]\), open at \(-4\) and closed at \(6\); its length is \(6-(-4)=10\).

Q: State De Morgan's laws.

\((A\cup B)'=A'\cap B'\) and \((A\cap B)'=A'\cup B'\): the complement of a union is the intersection of complements and vice versa.

Q: Distinguish a finite from an infinite set.

A set is finite if it is empty or its elements can be counted to a definite natural number \(n(S)\); otherwise it is infinite, e.g. \(\mathbb{N}\).

Q: Define disjoint sets.

Sets \(A\) and \(B\) are disjoint if \(A\cap B=\varphi\), i.e. they have no element in common, e.g. \(\{2,4,6\}\) and \(\{1,3,5\}\).

Q: List the complement laws for a subset \(A\) of \(U\).

\(A\cup A'=U\), \(A\cap A'=\varphi\), \((A')'=A\), \(\varphi'=U\) and \(U'=\varphi\).

Q: Prove that \(A\cup(A\cap B)=A\) (absorption law).

Every element of \(A\cap B\) already lies in \(A\), so the union adds nothing new; hence \(A\cup(A\cap B)=A\).

Class 11 Mathematics · Chapter 1

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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?

Flashcards & interactive practice