Q.Which of the following sets are finite or infinite

A set is finite if we can count its elements and reach an end; it is infinite if the counting never terminates.

(i) The set of months of a year

There are exactly 12 months: January through December. The count stops at 12.

This set is finite.

(ii) $\{1, 2, 3, \ldots\}$

The natural numbers continue without bound—there is no largest natural number.

This set is infinite.

(iii) $\{1, 2, 3, \ldots, 99, 100\}$

The list explicitly ends at 100; we have exactly 100 elements.

This set is finite.

(iv) The set of positive integers greater than 100

This is $\{101, 102, 103, \ldots\}$, which has no upper limit.

This set is infinite.

(v) The set of prime numbers less than 99

Primes less than 99 form a definite list (2, 3, 5, 7, …, 97)—there are exactly 25 of them.

This set is finite.

Understanding Finite vs. Infinite Sets

The distinction between finite and infinite sets rests on a simple question: can you finish counting the elements?

A finite set has a definite number of elements — you can list them all, and the list ends. We can assign a natural number $n$ to represent how many elements it contains (its cardinality).

An infinite set has no end. No matter how long you count, there are always more elements. The natural numbers themselves are the classic example: you can always add one more.

The key is to look at the defining property of each set and ask whether that property admits a last element or continues forever.

Let's examine each set:

1. The set of months of a year

   A year has exactly twelve months: January, February, March, …, December. We can list them all, and the list is complete. This set has cardinality $12$.

   This is a finite set.

2. $\{1, 2, 3, \ldots\}$

   The ellipsis "$\ldots$" here means "and so on, without end." This is the set of all natural numbers (positive integers). For any number you name, there's always a larger one: if you say $n$, I can say $n+1$. The counting never stops.

   This is an infinite set.

   Watch out

   The notation matters! An ellipsis with no upper bound (like here) signals infinity. An ellipsis with an upper bound (like in part iii) signals a finite list that's just abbreviated for convenience.

3. $\{1, 2, 3, \ldots, 99, 100\}$

   Here the ellipsis is bounded: it means "continue this pattern up to and including 100." We're listing the first hundred natural numbers. Though we've abbreviated the middle, every element is accounted for. The cardinality is $100$.

   This is a finite set.

4. The set of positive integers greater than 100

   This set is $\{101, 102, 103, \ldots\}$. It has a starting point (101) but no ending point. For any integer $n > 100$, the integer $n+1$ is also in the set and is larger. The pattern continues indefinitely.

   This is an infinite set.

5. The set of prime numbers less than 99

   A prime number is a natural number greater than $1$ whose only divisors are $1$ and itself. We need primes strictly less than $99$:

There are exactly $25$ such primes. The condition "less than 99" imposes an upper bound, so we can list them all.

This is a finite set.

Method: Cardinality Comparison with Natural Numbers

We determine whether a set is finite or infinite by checking if its elements can be counted and exhausted (finite) or if they go on without end (infinite). The natural numbers $\{1, 2, 3, \dots\}$ are the standard infinite set for comparison.

Steps

1. Identify the pattern or rule that defines the set’s elements.
2. Check if there is a largest or last element — if yes, the set is finite; if no, it is infinite.
3. Compare with natural numbers — if the set can be put into a one-to-one correspondence with a subset of $\{1, 2, \dots, n\}$ for some $n$, it is finite. If it matches $\{1, 2, 3, \dots\}$ without end, it is infinite.

Applying the method

(i) The set of months of a year

• Rule: 12 specific months (January to December).
• There is a last element (December).
• Finite — exactly 12 elements.

(ii) $\{1, 2, 3, \dots\}$

• Rule: all natural numbers, no end.
• No largest element; continues forever.
• Infinite — matches the natural numbers themselves.

(iii) $\{1, 2, 3, \dots, 99, 100\}$

• Rule: natural numbers from 1 to 100.
• Last element is 100.
• Finite — exactly 100 elements.

(iv) The set of positive integers greater than 100

• Rule: $\{101, 102, 103, \dots\}$ — no upper bound.
• No largest element.
• Infinite — can be paired with $\{1, 2, 3, \dots\}$ via $n \mapsto n+100$.

(v) The set of prime numbers less than 99

• Rule: primes like 2, 3, 5, 7, …, up to 97.
• There is a largest prime less than 99 (which is 97).
• Finite — the list stops at 97.

Final Answer

Here are the common mistakes students make when classifying sets as finite or infinite, along with how to avoid each — tailored for Indian exam accuracy.

✗ Mistake 1: Confusing "infinite" with "very large but finite"

Example:

Thinking $\{1, 2, 3, \dots, 99, 100\}$ is infinite because it has "many" elements.

Why it happens:

Students see the ellipsis (...) and assume it goes on forever, ignoring the explicit endpoint 100.

How to avoid:

• Look for the last term. If a set is written as $\{a, a+1, \dots, b\}$ with a clear last number $b$, it is finite.
• Count the elements: here, exactly 100 elements → finite.

✓ Correct: Finite (100 elements)

✗ Mistake 2: Misreading the ellipsis ... in $\{1, 2, 3, \dots\}$

Example:

Calling $\{1, 2, 3, \dots\}$ finite because "it starts at 1 and ends at 3".

Why it happens:

Students treat ... as decorative, not as "continues indefinitely".

How to avoid:

• The three dots without an endpoint mean "goes on forever".
• If there is no last number listed, the set is infinite.

✓ Correct: Infinite (set of all natural numbers)

✗ Mistake 3: Thinking "greater than 100" includes 100

Example:

Saying "The set of positive integers greater than 100" is finite because "it stops at some large number".

Why it happens:

Confusion between "greater than" and "greater than or equal to". Also, students imagine a largest integer.

How to avoid:

• "Greater than 100" means $101, 102, 103, \dots$ — no upper bound.
• There is no largest integer — so the set is infinite.

✓ Correct: Infinite

✗ Mistake 4: Forgetting that "prime numbers less than 99" are finite

Example:

Calling "The set of prime numbers less than 99" infinite because "there are infinitely many primes".

Why it happens:

Students recall the theorem "there are infinitely many primes" but ignore the restriction "less than 99".

How to avoid:

• Always check the condition. "Less than 99" means only primes from 2 up to 97.
• Count them: there are 25 primes less than 100 → finite.

✓ Correct: Finite (25 elements)

✗ Mistake 5: Confusing "set of months" with "set of days in a month"

Example:

Thinking "The set of months of a year" is infinite because "months keep repeating every year".

Why it happens:

Students think of time as cyclic and endless, but the set is about distinct month names, not occurrences.

How to avoid:

• A set collects distinct objects. The months are: Jan, Feb, ..., Dec — exactly 12.
• Repetition over years does not add new elements to the set.

✓ Correct: Finite (12 elements)

📋 Quick Summary Table

✓ Final Exam Tip

Always ask: Does this set have a last element?

• If yes → finite
• If no → infinite

This single question will save you from most mistakes in set membership problems.