Given that N = \1, 2, 3, …, 100\. Then write (i) the subset of N whose elements are even numbers. (ii) the sub

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N = \{1, 2, 3, \ldots, 100\}

(i) the subset of

N

whose elements are even numbers.

(ii) the subset of

N

whose elements are perfect square numbers.

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Concept understanding — Subset Listing

Subset Listing: A First Look

Let's build this from the ground up — no jargon, just intuition first.

1. The Intuition: What does "subset" mean?

Imagine you have a set — a collection of distinct objects. For example:

Set A = {apple, banana, cherry}

Now, a subset is simply a selection of some (or all, or none) of these objects, taken from the original set.

You could pick all three → {apple, banana, cherry}
You could pick just two → {apple, banana}
You could pick just one → {cherry}
You could pick none → {} (the empty set)

Each of these is a subset of the original set.

2. The Precise Definition

Definition: A set $B$ is a subset of a set $A$ if every element of $B$ is also an element of $A$ .

We write this as:

B \subseteq A

If $B$ is not a subset of $A$ , we write:

B \not\subseteq A

Key points to remember:

Every set is a subset of itself.

Example: {apple, banana} ⊆ {apple, banana}
The empty set $\emptyset$ (or $\{\}$ ) is a subset of every set.

Why? Because it has no elements, so there's nothing to violate the condition.
If $B$ is a subset of $A$ but $B \neq A$ , we call $B$ a proper subset.

Notation: $B \subset A$ (some books use $\subsetneq$ )

3. How to "list" all subsets

Subset listing means writing down every possible subset of a given set.

Example: Set $S = \{a, b\}$

All subsets:

$\emptyset$ (empty set)
$\{a\}$
$\{b\}$
$\{a, b\}$ (the set itself)

So the list of all subsets is:

\{ \emptyset, \{a\}, \{b\}, \{a, b\} \}

How many subsets does a set have?

If a set has $n$ elements, it has exactly $2^n$ subsets.

$n = 0$ → $2^0 = 1$ subset (just the empty set)
$n = 1$ → $2^1 = 2$ subsets
$n = 2$ → $2^2 = 4$ subsets (as above)
$n = 3$ → $2^3 = 8$ subsets

Why $2^n$ ?

For each element, you have 2 choices: include it or exclude it. Multiply these choices: $2 \times 2 \times \dots \times 2$ ( $n$ times) = $2^n$ .

4. A systematic way to list subsets

For a set with $n$ elements, you can use a binary counting method:

Label each element with a position (1st, 2nd, 3rd, ...)
Count from $0$ to $2^n - 1$ in binary
Each binary number tells you which elements to include (1 = include, 0 = exclude)

Example: $S = \{a, b, c\}$ (3 elements)

Binary	Subset
000	$\emptyset$
001	$\{c\}$
010	$\{b\}$
011	$\{b, c\}$
100	$\{a\}$

Why this formula?

Okay, let's break down Subset Listing from the ground up. The core idea is simple: given a set, how do we systematically list all its subsets, and why does the formula $2^n$ work?

1. The Core Question

Imagine you have a set with $n$ elements, like $S = \{a, b, c\}$ (so $n=3$ ). A subset is any collection of elements from $S$ , including the empty set $\{\}$ and the set itself $\{a, b, c\}$ .

The key formula is:

Total number of subsets of a set with $n$ elements = $\mathbf{2^n}$

Let's see why this is true, not just memorize it.

2. The "Decision" or "Binary Choice" Reasoning

The most intuitive derivation comes from thinking about each element individually.

For each element in the original set, when building a subset, you have exactly two choices:

Include the element in the subset.
Exclude the element from the subset.

This is a fundamental, independent decision for every element.

Example with $S = \{a, b, c\}$

For element $a$ : Choose IN or OUT. (2 choices)
For element $b$ : Choose IN or OUT. (2 choices)
For element $c$ : Choose IN or OUT. (2 choices)

Since these choices are independent (choosing for $a$ doesn't affect the choice for $b$ ), the total number of distinct combinations of choices is the product of the number of choices for each element:

2 \times 2 \times 2 = 2^3 = 8

This directly gives the 8 subsets of $\{a, b, c\}$ :

$\{\}$ (all OUT)
$\{a\}$ (a IN, b OUT, c OUT)
$\{b\}$
$\{c\}$
$\{a, b\}$
$\{a, c\}$
$\{b, c\}$
$\{a, b, c\}$ (all IN)

3. The General Formula (Derivation)

For a set with $n$ elements, you have $n$ independent binary decisions. Therefore:

\text{Total subsets} = \underbrace{2 \times 2 \times \dots \times 2}_{n \text{ times}} = 2^n

This is the fundamental reason the formula holds. It's not a coincidence; it's a direct consequence of the counting principle for independent events.

4. Why This Matters for Exams

Don't just memorize $2^n$ . If a question asks "How many subsets does a set with 5 elements have?", you can instantly say $2^5 = 32$ . But if they ask why, you now have the reasoning. …

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We are asked to identify two subsets from the set $N = \{1, 2, 3, \ldots, 100\}$ . The subset of even numbers is $\{2, 4, 6, \ldots, 100\}$ , and the subset of perfect square numbers is $\{1, 4, 9, 16, 25, 36, 49, 64, 81, 100\}$ .

When we talk about a "subset," we are looking for a collection of elements that are all part of a larger, original set. The key idea is that every element in the subset must also be an element of the original set. The problem asks us to form subsets based on specific properties: being an even number or being a perfect square number. To do this, we simply examine each element in the given set $N$ and check if it satisfies the stated property. If it does, we include it in our new subset.

Let's break this down for each part.

(i) The subset of $N$ whose elements are even numbers.

Understanding "even numbers": An even number is any integer that is exactly divisible by 2. This means it can be written in the form $2k$ , where $k$ is an integer. Examples are $2, 4, 6, 0, -2$ , etc.
Identifying even numbers in $N$ : The set $N$ $N$ is given as $\{1, 2, 3, \ldots, 100\}$ ${1, 2, 3, \dots, 100}$ . We need to find all numbers in this range that are even.
- The smallest number in $N$ is $1$ . It is not even.
- The next number is $2$ . It is even ( $2 = 2 \times 1$ ).
- The next even number is $4$ ( $4 = 2 \times 2$ ).
- This pattern continues, adding $2$ each time.
- The largest number in $N$ is $100$ . Since $100 \div 2 = 50$ (with no remainder), $100$ is an even number.
Forming the subset: We collect all these even numbers from $N$ . Let's call this subset $E$ . $E = \{x \in N \mid x \text{ is an even number}\}$ $E = \{2, 4, 6, \ldots, 100\}$

(ii) The subset of $N$ whose elements are perfect square numbers.

Understanding "perfect square numbers": A perfect square number is an integer that can be expressed as the product of an integer with itself. In other words, it is the square of an integer. For example, $1$ is a perfect square because $1 = 1^2$ , $4$ is a perfect square because $4 = 2^2$ , $9$ is a perfect square because $9 = 3^2$ , and so on. …

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Q.Given that N={1,2,3,…,100}N = \{1, 2, 3, \ldots, 100\}N={1,2,3,…,100}. Then write

(i) The subset of NNN whose elements are even numbers.

(ii) The subset of NNN whose elements are perfect square numbers.

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Q.Given that $N = \{1, 2, 3, \ldots, 100\}$ . Then write

(i) The subset of $N$ whose elements are even numbers.

(ii) The subset of $N$ whose elements are perfect square numbers.

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