Business Mathematics and Basic Statistics · Class 11 Commerce
Ch 5Theory of Sets — Introduction — Class 11 Business Mathematics and Basic Statistics, concept-first.
A set is a well-defined collection of distinct objects. "Well-defined" means that, given any object, we can say with certainty whether it belongs to the collection or not — "the set of vowels in the English alphabet" is well-defined (we can check any letter against it), but "the set of intelligent students in a class"…
Key concepts
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Set Definition and Representation
A set is a well-defined collection of distinct objects (its elements), written using curly-brace notation, with membership shown by and non-membership by .
Most relevant Q&A
In previous exams
How often this chapter’s concepts have been examined — real appearance data, never estimated.
Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
What is a Set? — Definition and Notation
A set is a well-defined collection of distinct objects. "Well-defined" means that, given any object, we can say with certainty whether it belongs to the collection or not — "the set of vowels in the E…
Representing a Set — Roster Form and Set-Builder Form
A set can be written down in two standard ways.
Special Types of Sets
A few particular kinds of sets come up so often that they are given their own names.
Cartesian Product of Two Sets
Given two non-empty sets and , the Cartesian product is the set of all possible ordered pairs where the first entry comes from and the second entry comes from :
Cardinality — Counting the Elements of a Set
The cardinality of a finite set is the number of distinct elements it contains, written or . For example, if , then .
Subset, Superset and Proper Subset
A set is called a subset of a set — written — if every element of is also an element of . Formally, When , we also say is a superset of , written .
Exercises
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- Q7Write the set of all prime numbers less than 20 in roster form.Free
- Q8Write the set $\{5, 10, 15, 20, 25\}$ in set-builder form.Free
- Q9Let $P = \{a, b, c\}$, $Q = \{a, b, c, d, e\}$ and $R = \{a, b, c\}$. State, giving reasons, which of the following are true: (i) $P \subset…Preview
- Q10If $X = \{p, q, r, s\}$ and $Y = \{1, 2\}$, find $n(X)$, $n(Y)$ and $n(X \times Y)$. Also list $X \times Y$.Preview
- Q11Classify each of the following as empty, singleton or finite: (i) $\{x : x \in \mathbb{N}, 3 < x < 4\}$ (ii) the set of even prime numbers (…Preview
- Q12A universal set is given as $U = \{1, 2, 3, \dots, 10\}$. Let $A = \{2, 4, 6, 8, 10\}$ be the set of even numbers in $U$. (i) Is $A \subsete…Preview
More questions
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- Example 1Write the set $A = \{2, 4, 6, 8, 10\}$ in set-builder form.Free
- Example 2Write the set $B = \{x : x \in \mathbb{N}, x^2 < 30\}$ in roster form.Free
- Example 3Classify each of the following as empty, singleton, finite, or universal (taking $\mathbb{N}$ as the universal set for this problem): (i) $\…Preview
- Example 4If $A = \{1, 2\}$ and $B = \{3, 4, 5\}$, find $A \times B$ and $|A \times B|$.Preview
- Example 5Show that $\mathbb{N} \subseteq \mathbb{W}$ and $\mathbb{N} \subset \mathbb{W}$, where $\mathbb{N} = \{1, 2, 3, \dots\}$ and $\mathbb{W} = \…Preview
- Example 6Let $A = \{1, 2\}$ and $B = \{1, 2, 3\}$. Verify that $A \subset B \Rightarrow A \subseteq B$. Then, using $C = \{1, 2\}$ and $D = \{1, 2\}$…Preview