Mathematics: Class 11 — Chapter 1: SETS

• Objects, elements, and members of a set are synonymous terms.
• Sets are usually denoted by capital letters: $A, B, C, X, Y, Z$, etc.
• The elements of a set are represented by small letters: $a, b, c, x, y, z$, etc.
• If $a$ is an element of a set $A$, we say that "$a$ belongs to $A$" and write $a \in A$.
• If $b$ is not an element of a set $A$, we write $b \notin A$ (read as "$b$ does not belong to $A$").

The following symbols are standard representations for specific sets of numbers:

There are two primary methods of representing a set:

1. Roster or Tabular Form: All the elements of the set are listed, separated by commas, and enclosed within braces $\{\}$.
   • Rule of Order: The order in which elements are listed is immaterial. For example, $\{1, 3, 7\}$ is the same as $\{3, 7, 1\}$.
   • Rule of Repetition: Elements are not generally repeated; all elements are taken as distinct. For example, the set of letters in "SCHOOL" is $\{S, C, H, O, L\}$.
2. Set-Builder Form: All elements of the set possess a single common property which is not possessed by any element outside the set.
   • We describe the element using a symbol (usually $x$), followed by a colon "$:$" (which stands for "such that"), and then write the characteristic property.
   • Example: $V = \{x : x \text{ is a vowel in the English alphabet}\}$.

Mathematical Problem Statement: Write the solution set of the equation $x^2 + x - 2 = 0$ in roster form.

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Step-by-Step Algorithmic Solution Trace:

1. Factorize the quadratic equation: $x^2 + x - 2 = 0 \implies (x - 1)(x + 2) = 0$.
2. Solve for $x$: $x = 1$ or $x = -2$.
3. Represent the solutions as elements of a set enclosed in braces.

Final Evaluation Output: $\{1, -2\}$

Mathematical Problem Statement: Write the set $\{x : x \text{ is a positive integer and } x^2 < 40\}$ in the roster form.

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Step-by-Step Algorithmic Solution Trace:

1. Identify positive integers: $1, 2, 3, 4, 5, 6, 7, \dots$
2. Evaluate their squares: $1^2=1, 2^2=4, 3^2=9, 4^2=16, 5^2=25, 6^2=36, 7^2=49$.
3. Select those whose squares are strictly less than $40$: $1, 2, 3, 4, 5, 6$.

Final Evaluation Output: $\{1, 2, 3, 4, 5, 6\}$

Mathematical Problem Statement: Write the set $\left\{\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \frac{6}{7}\right\}$ in the set-builder form.

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Step-by-Step Algorithmic Solution Trace:

1. Observe the pattern of the elements: each fraction has a numerator $n$ and a denominator $n+1$.
2. Identify the domain of $n$: $n$ is a natural number starting from $1$ up to $6$ ($1 \le n \le 6$).
3. Formulate the characteristic property: $x = \frac{n}{n+1}$ where $n \in \mathbf{N}$ and $1 \le n \le 6$.

Final Evaluation Output: $\left\{x : x = \frac{n}{n+1}, \text{ where } n \in \mathbf{N} \text{ and } 1 \le n \le 6\right\}$

A set which does not contain any element is called the empty set, null set, or void set.

• Notation: Denoted by the Greek symbol $\phi$ or empty braces $\{\}$.
• Examples:
  • $A = \{x : 1 < x < 2, x \text{ is a natural number}\} = \phi$
  • $B = \{x : x^2 - 2 = 0 \text{ and } x \text{ is a rational number}\} = \phi$

• A set which is empty or consists of a definite number of elements is called a finite set.
• Otherwise, the set is called an infinite set.
• Cardinality: For a finite set $S$, the number of distinct elements is denoted by $n(S)$.
• Representation of Infinite Sets: Represented in roster form by writing a few elements to show the structure, followed by three dots (ellipsis). For example, the set of natural numbers is $\{1, 2, 3, \dots\}$.
• Note: All infinite sets cannot be described in roster form. For example, the set of real numbers $\mathbf{R}$ cannot be described this way because its elements do not follow any particular pattern.

Two sets $A$ and $B$ are said to be equal if they have exactly the same elements, and we write $A = B$. Otherwise, they are unequal ($A \neq B$).

• Mathematical Definition: $A = B$ if every element of $A$ is in $B$ and every element of $B$ is in $A$.
• Note: Repetition or reordering of elements does not change a set. For example, $\{1, 2, 3\} = \{2, 2, 1, 3, 3\}$.

Mathematical Problem Statement: State which of the following sets are finite or infinite:
(i) $\{x : x \in \mathbf{N} \text{ and } (x - 1)(x - 2) = 0\}$
(ii) $\{x : x \in \mathbf{N} \text{ and } x^2 = 4\}$
(iii) $\{x : x \in \mathbf{N} \text{ and } 2x - 1 = 0\}$
(iv) $\{x : x \in \mathbf{N} \text{ and } x \text{ is prime}\}$

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Step-by-Step Algorithmic Solution Trace:

1. For (i), the solutions are $x = 1, 2$. Both are natural numbers. The set is $\{1, 2\}$, which is finite.
2. For (ii), $x^2 = 4 \implies x = 2$ (since $x \in \mathbf{N}$). The set is $\{2\}$, which is finite.
3. For (iii), $2x - 1 = 0 \implies x = 1/2$, which is not a natural number. The set is $\phi$, which is finite.
4. For (iv), the set of prime numbers is $\{2, 3, 5, 7, 11, \dots\}$, which is infinite.

Final Evaluation Output: (i) Finite, (ii) Finite, (iii) Finite, (iv) Infinite

A set $A$ is said to be a subset of a set $B$ if every element of $A$ is also an element of $B$. We write $A \subset B$.

• Mathematical Definition: $A \subset B \iff (a \in A \implies a \in B)$.
• If $A$ is not a subset of $B$, we write $A \not\subset B$.
• Proper Subset & Superset: If $A \subset B$ and $A \neq B$, then $A$ is called a proper subset of $B$, and $B$ is called the superset of $A$.
• Singleton Set: A set containing only a single element, denoted as $\{a\}$.
• Important Theorems:
  1. Every set is a subset of itself: $A \subset A$.
  2. The empty set is a subset of every set: $\phi \subset A$.

The relationships between standard number systems can be expressed using subset notation:

$$\mathbf{N} \subset \mathbf{Z} \subset \mathbf{Q} \subset \mathbf{R}$$

Additionally, if $\mathbf{T}$ is the set of all irrational numbers, then:

$$\mathbf{T} \subset \mathbf{R}, \quad \mathbf{N} \not\subset \mathbf{T}$$

Let $a, b \in \mathbf{R}$ and $a < b$. We define the following types of intervals:

The number $(b - a)$ is called the length of any of these intervals.

The Universal Set is a basic set containing all possible objects under consideration for a particular context. It is denoted by $U$, and all other sets are treated as its subsets.

• Example: In the study of number systems, the universal set can be the set of real numbers $\mathbf{R}$.
• Example: In human population studies, the universal set consists of all the people in the world.

Relationships between sets can be represented geometrically using Venn diagrams, named after the English logician John Venn (1834-1883).

• The universal set $U$ is represented by a rectangle.
• Subsets of $U$ are represented by closed curves, usually circles, inside the rectangle.
• Elements of the sets are written inside their respective geometric shapes.

The union of two sets $A$ and $B$ is the set which consists of all elements of $A$ and all elements of $B$, with common elements taken only once. Denoted by $A \cup B$.

$$A \cup B = \{x : x \in A \text{ or } x \in B\}$$

Mathematical Problem Statement: Let $A = \{2, 4, 6, 8\}$ and $B = \{6, 8, 10, 12\}$. Find $A \cup B$.

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Step-by-Step Algorithmic Solution Trace:

1. List all elements of $A$: $2, 4, 6, 8$.
2. List all elements of $B$: $6, 8, 10, 12$.
3. Combine the lists, ensuring common elements ($6, 8$) are written only once.

Final Evaluation Output: $\{2, 4, 6, 8, 10, 12\}$

The intersection of two sets $A$ and $B$ is the set of all elements which are common to both $A$ and $B$. Denoted by $A \cap B$.

$$A \cap B = \{x : x \in A \text{ and } x \in B\}$$

• Disjoint Sets: If $A$ and $B$ have no common elements, i.e., $A \cap B = \phi$, they are called disjoint sets.

Mathematical Problem Statement: Let $A = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ and $B = \{2, 3, 5, 7\}$. Find $A \cap B$ and show that $A \cap B = B$.

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Step-by-Step Algorithmic Solution Trace:

1. Identify elements common to both $A$ and $B$: $2, 3, 5, 7$.
2. Construct the intersection set: $A \cap B = \{2, 3, 5, 7\}$.
3. Observe that this set is identical to $B$.

Final Evaluation Output: $A \cap B = \{2, 3, 5, 7\} = B$

The difference of two sets $A$ and $B$ (written as $A - B$) is the set of elements which belong to $A$ but not to $B$.

$$A - B = \{x : x \in A \text{ and } x \notin B\}$$

• Note: $A - B \neq B - A$ in general.
• Disjoint Property: The three sets $(A - B)$, $(A \cap B)$, and $(B - A)$ are mutually disjoint sets.

Mathematical Problem Statement: Let $A = \{1, 2, 3, 4, 5, 6\}$ and $B = \{2, 4, 6, 8\}$. Find $A - B$ and $B - A$.

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Step-by-Step Algorithmic Solution Trace:

1. To find $A - B$, remove all elements of $B$ from $A$: remove $2, 4, 6$ from $\{1, 2, 3, 4, 5, 6\} \implies \{1, 3, 5\}$.
2. To find $B - A$, remove all elements of $A$ from $B$: remove $2, 4, 6$ from $\{2, 4, 6, 8\} \implies \{8\}$.

Final Evaluation Output: $A - B = \{1, 3, 5\}$ and $B - A = \{8\}$

Let $U$ be the universal set and $A$ be a subset of $U$. The complement of A (denoted by $A'$) is the set of all elements of $U$ which are not elements of $A$.

$$A' = U - A = \{x : x \in U \text{ and } x \notin A\}$$

Mathematical Problem Statement: Let $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ and $A = \{1, 3, 5, 7, 9\}$. Find $A'$.

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Step-by-Step Algorithmic Solution Trace:

1. Identify elements in $U$ that are not in $A$.
2. The elements are $2, 4, 6, 8, 10$.
3. Construct the complement set: $A' = \{2, 4, 6, 8, 10\}$.

Final Evaluation Output: $A' = \{2, 4, 6, 8, 10\}$

Mathematical Problem Statement: Let $U = \{1, 2, 3, 4, 5, 6\}$, $A = \{2, 3\}$ and $B = \{3, 4, 5\}$. Verify that $(A \cup B)' = A' \cap B'$.

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Step-by-Step Algorithmic Solution Trace:

1. Compute $A \cup B = \{2, 3, 4, 5\}$.
2. Find the complement of $A \cup B$: $(A \cup B)' = U - \{2, 3, 4, 5\} = \{1, 6\}$.
3. Compute $A' = U - A = \{1, 4, 5, 6\}$.
4. Compute $B' = U - B = \{1, 2, 6\}$.
5. Find the intersection of $A'$ and $B'$: $A' \cap B' = \{1, 4, 5, 6\} \cap \{1, 2, 6\} = \{1, 6\}$.
6. Compare the results: $(A \cup B)' = A' \cap B' = \{1, 6\}$.

Final Evaluation Output: Verified that $(A \cup B)' = A' \cap B' = \{1, 6\}$