State which of the following statements are true and which are false. Justify your answer. (i) 35 ∈ \x mid x h

NCERT Exemplar · Q4

Q. State which of the following statements are true and which are false. Justify your answer.

(i)

35 \in \{x \mid x \text{ has exactly four positive factors}\}

(ii)

128 \in \{y \mid \text{the sum of all the positive factors of } y \text{ is } 2y\}

(iii)

3 \notin \{x \mid x^4 - 5x^3 + 2x^2 - 112x + 6 = 0\}

(iv)

496 \notin \{y \mid \text{the sum of all the positive factors of } y \text{ is } 2y\}

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Concept understanding — Set Membership

Set Membership

The idea in plain words

Every set is defined by exactly one question: "does this object belong to the set, or not?" That yes/no relationship between an object and a set is called membership. If an object is in the set, it is a member (or element) of the set; if it isn't, it simply is not.

A set is only "well-defined" if this question always has a clear answer for every possible object — that's what makes membership testable.

The notation

For a set $A$ and an object $x$ :

$x \in A$ reads " $x$ belongs to $A$ " or " $x$ is an element of $A$ " — TRUE membership.
$x \notin A$ reads " $x$ does not belong to $A$ " — FALSE membership.

Example. Let $A = \{2, 4, 6, 8\}$ .

4 \in A \qquad \text{(4 is listed inside A)}

5 \notin A \qquad \text{(5 is not listed inside A)}

Testing membership: roster form vs. set-builder form

Roster form — just look for the object in the list.

B = \{1, 3, 5, 7\}, \quad 3 \in B, \quad 4 \notin B

Set-builder form — plug the candidate into the defining rule and check if it's satisfied.

C = \{x \mid x \text{ is a prime number less than } 10\}

Is $7 \in C$ ? Check: is 7 prime and less than 10? Yes → $7 \in C$ .

Is $9 \in C$ ? Check: 9 is less than 10 but not prime ( $9 = 3 \times 3$ ) → $9 \notin C$ .

Properties every student must know

Each element is either in or out — never "partly in." Membership is binary, not a matter of degree.
Repetition doesn't affect membership. $\{1, 1, 2\} = \{1, 2\}$ — asking "is 1 a member?" gives the same YES either way.
Order never affects membership. $2 \in \{1, 2, 3\}$ is exactly the same fact as $2 \in \{3, 2, 1\}$ .
Nothing belongs to the empty set. For any $x$ , $x \notin \emptyset$ — there is nothing inside to belong to.
A set can be an element of another set. If $D = \{1, \{2, 3\}\}$ , then $1 \in D$ is true, and $\{2,3\} \in D$ is true, but $2 \in D$ is false — 2 is not directly listed in $D$ ; it is only inside the set that is listed.

A common slip

Students often confuse $x \in A$ (membership: is the object present?) with $B \subseteq A$ (subset: is every element of $B$ also in $A$ ?). The two symbols compare different kinds of things:

$\in$ / $\notin$ compares an element to a set.
$\subseteq$ / $\not\subseteq$ compares a set to a set. …

Why this formula?

Let's break down the definition of a set — not as a formula to memorise, but as a fundamental idea that underpins all of mathematics.

1. What is a Set? (The Core Idea)

A set is a well-defined collection of distinct objects.

The "why" here is about clarity and precision — we need to know exactly what belongs and what does not.

Well-defined: For any object, we can say yes or no — no ambiguity.
Distinct: No duplicates — each object appears only once.

Why? Because if we couldn't decide membership, we couldn't do any logical operations. Sets are the building blocks of all mathematical structures.

2. The Key "Formula": Set-Builder Notation

The most common way to define a set is:

S = \{ x \mid P(x) \}

This reads: "S is the set of all objects $x$ such that property $P(x)$ is true."

Why does this work?

$x$ is a placeholder for any object.
$P(x)$ is a logical condition (a predicate) that is either true or false for each $x$ .
The vertical bar $\mid$ means "such that".

Example:

A = \{ n \mid n \in \mathbb{N}, n \text{ is even} \}

Here, $P(n)$ is " $n$ is a natural number and $n$ is even".

Only those $n$ that satisfy both conditions are included.

Why this form? It avoids listing infinitely many elements. It gives a rule — a decision procedure — for membership.

3. The Two Fundamental Properties (Axioms)

Every set definition relies on two intuitive truths:

(a) Extensionality — Two sets are equal if they have the same elements.

A = B \iff (\forall x)(x \in A \iff x \in B)

Why? A set is completely determined by its members. There is no other hidden property.

If you know what's inside, you know the set.

(b) Membership — The only relation is $\in$ (belongs to).

x \in S \quad \text{or} \quad x \notin S

Why? Because a set is just a container. The only question we can ask is: "Is this object inside?"

4. Why Can't We Just List Everything?

For small sets, listing works:

\{1, 2, 3\}

But for infinite sets (like all natural numbers), listing is impossible.

Set-builder notation solves this by giving a rule instead of a list.

Example:

\mathbb{N} = \{ n \mid n \text{ is a positive integer} \}

This is not a formula to memorise — it's a definition by property.

5. The "Empty Set" — Why It Exists

The empty set $\emptyset$ (or $\{\}$ ) is the set with no elements. …

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We evaluate four statements involving set membership and number properties. Statement (i) is True because 35 has exactly four factors. Statement (ii) is False because 128 is not a perfect number. Statement (iii) is True because 3 is not a root of the given polynomial. Statement (iv) is False because 496 is a perfect number.

Let's analyze each statement by understanding the properties of the numbers involved and the definitions of the sets.

Statement (i): $35 \in \{x \mid x \text{ has exactly four positive factors}\}$

Concept: The number of positive factors of an integer.

To determine if a number belongs to this set, we need to find its prime factorization. If a number $n$ has the prime factorization $n = p_1^{a_1} p_2^{a_2} \dots p_k^{a_k}$ , then the total number of positive factors is given by the product of one more than each exponent: $(a_1+1)(a_2+1)\dots(a_k+1)$ .

Find the prime factorization of 35:

$35 = 5^1 \times 7^1$ .
Calculate the number of factors:

Using the formula, the number of factors of 35 is $(1+1)(1+1) = 2 \times 2 = 4$ .

The factors are indeed $1, 5, 7, 35$ .
Evaluate the statement:

Since 35 has exactly four positive factors, it satisfies the condition for membership in the set.

Therefore, the statement $35 \in \{x \mid x \text{ has exactly four positive factors}\}$ is True.

Statement (ii): $128 \in \{y \mid \text{the sum of all the positive factors of } y \text{ is } 2y\}$

Concept: Perfect numbers.

A positive integer $y$ is called a perfect number if the sum of its proper positive divisors (divisors excluding $y$ itself) is equal to $y$ . An equivalent definition, and often easier to work with, is that the sum of all its positive divisors (including $y$ ) is equal to $2y$ . We need to check if 128 satisfies this condition.

n = p_1^{a_1} p_2^{a_2} \dots p_k^{a_k}

Find the prime factorization of 128:

$128 = 2^7$ .
Calculate the sum of factors of 128:

Using the formula for the sum of factors:

$\sigma(128) = \frac{2^{7+1}-1}{2-1} = \frac{2^8-1}{1} = 256-1 = 255$ .
Check the perfect number condition:

We need to see if $\sigma(128) = 2 \times 128$ .

$255 = 2 \times 128$

$255 = 256$ .

This is false. $255 \neq 256$ .
Evaluate the statement:

Since the sum of the factors of 128 is not equal to $2 \times 128$ , 128 is not a perfect number.

Therefore, the statement $128 \in \{y \mid \text{the sum of all the positive factors of } y \text{ is } 2y\}$ is False.

Statement (iii): $3 \notin \{x \mid x^4 - 5x^3 + 2x^2 - 112x + 6 = 0\}$

Concept: Roots of a polynomial equation.

A number $a$ is a root of a polynomial equation $P(x) = 0$ if substituting $a$ for $x$ makes the equation true, i.e., $P(a) = 0$ . The set described is the set of all roots of the given polynomial equation. The statement claims that 3 is not a root.

Substitute $x=3$ into the polynomial:

Let $P(x) = x^4 - 5x^3 + 2x^2 - 112x + 6$ .

We need to calculate $P(3)$ :

$P(3) = (3)^4 - 5(3)^3 + 2(3)^2 - 112(3) + 6$
Evaluate the expression:

$P(3) = 81 - 5(27) + 2(9) - 336 + 6$

$P(3) = 81 - 135 + 18 - 336 + 6$

$P(3) = (81 + 18 + 6) - (135 + 336)$

$P(3) = 105 - 471$

$P(3) = -366$
Check if 3 is a root:

Since $P(3) = -366 \neq 0$ , $3$ is not a root of the equation $x^4 - 5x^3 + 2x^2 - 112x + 6 = 0$ .
Evaluate the statement:

The statement claims that $3 \notin \{x \mid x^4 - 5x^3 + 2x^2 - 112x + 6 = 0\}$ , which means 3 is not in the set of roots. Our calculation confirms this.

Therefore, the statement $3 \notin \{x \mid x^4 - 5x^3 + 2x^2 - 112x + 6 = 0\}$ is True.

Statement (iv): $496 \notin \{y \mid \text{the sum of all the positive factors of } y \text{ is } 2y\}$

Concept: Perfect numbers (same as part ii). …

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Q.State which of the following statements are true and which are false. Justify your answer.

Statement (i): 35∈{x∣x has exactly four positive factors}35 \in \{x \mid x \text{ has exactly four positive factors}\}35∈{x∣x has exactly four positive factors}

Statement (ii): 128∈{y∣the sum of all the positive factors of y is 2y}128 \in \{y \mid \text{the sum of all the positive factors of } y \text{ is } 2y\}128∈{y∣the sum of all the positive factors of y is 2y}

Statement (iii): 3∉{x∣x4−5x3+2x2−112x+6=0}3 \notin \{x \mid x^4 - 5x^3 + 2x^2 - 112x + 6 = 0\}3∈/{x∣x4−5x3+2x2−112x+6=0}

Statement (iv): 496∉{y∣the sum of all the positive factors of y is 2y}496 \notin \{y \mid \text{the sum of all the positive factors of } y \text{ is } 2y\}496∈/{y∣the sum of all the positive factors of y is 2y}

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Statement (i): $35 \in \{x \mid x \text{ has exactly four positive factors}\}$

Statement (ii): $128 \in \{y \mid \text{the sum of all the positive factors of } y \text{ is } 2y\}$

Statement (iii): $3 \notin \{x \mid x^4 - 5x^3 + 2x^2 - 112x + 6 = 0\}$

Statement (iv): $496 \notin \{y \mid \text{the sum of all the positive factors of } y \text{ is } 2y\}$

Unlock everything free for 14 days