Practice worksheet Set Theory- Class vii - Math with JM - home - Free Printable

Let's solve the Practice Worksheet on Set Theory step by step, with explanations.

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Question 1: Which of the following are sets? Justify your answer.

A set is a well-defined collection of distinct objects. For something to be a set, its elements must be clearly defined and identifiable (i.e., we should be able to determine whether an object belongs to the collection or not).

Let’s analyze each:

(a) The collection of all the days in a week beginning with the letter 'T'.

- Days starting with 'T': Tuesday, Thursday.
- This is a well-defined collection.
✔ Yes, it is a set.

(b) The collection of all difficult questions in the chapter on sets.

- "Difficult" is subjective — what one person finds difficult, another might not.
- Not well-defined.
✘ Not a set.

(c) The collection of girls in your class.

- Assuming we know who is in the class, this collection is clear and definite.
✔ Yes, it is a set.

(d) The collection of all rivers in India.

- Although large, the rivers in India can be identified and listed.
- Well-defined.
✔ Yes, it is a set.

(e) The collection of all active teachers in the school.

- "Active" may be ambiguous — does it mean currently teaching, employed, or working full-time?
- Could be interpreted differently.
- However, if we assume "active" means currently employed teachers, then it's well-defined.
✔ Yes, it is a set, assuming a clear definition of "active".

(f) The collection of all integers more than -3.

- Integers greater than -3: -2, -1, 0, 1, 2, 3, ...
- This is a well-defined mathematical collection.
✔ Yes, it is a set.

(g) The collection of all beautiful flowers in the park.

- "Beautiful" is subjective — depends on personal opinion.
- Not well-defined.
✘ Not a set.

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✔ Summary for Q1:

- Sets: (a), (c), (d), (e), (f)
- Not sets: (b), (g)

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Question 2:

Given:
- $ A = \{3, 5, 7, 9\} $
- $ B = \{2, 4, 6, 8, 10\} $
- $ C = \{12, 14, 18, 20, 24\} $
- $ D = \{21, 26, 31, 36\} $

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#### (a) State whether true or false:

(i) $ 13 \in C $
→ 13 is not in C → ✘ False

(ii) $ 4 \in A $
→ 4 is not in A → ✘ False

(iii) $ 9 \in A $
→ Yes, 9 is in A → ✔ True

(iv) $ 24 \in C $
→ Yes, 24 is in C → ✔ True

(v) $ 5 \in D $
→ 5 is not in D → ✘ False

(vi) $ 36 \in D $
→ Yes, 36 is in D → ✔ True

(vii) $ 20 \in C $
→ Yes, 20 is in C → ✔ True

(viii) $ 9 \in A $
→ Already confirmed → ✔ True

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#### (b) Fill in the blanks:

We use:
- $ \in $: element belongs to
- $ \notin $: element does not belong to
- $ \subset $: subset
- $ \subseteq $: subset or equal
- $ \not\subset $: not a subset

(i) $ 3 \in \_\_\_ $
→ 3 is in A → $ 3 \in A $

(ii) $ 4 \_\_\_ B $
→ 4 is in B → $ 4 \in B $

(iii) $ 26 \_\_\_ C $
→ 26 is not in C → $ 26 \notin C $

(iv) $ B \_\_\_ \_\_\_ $
→ We need to find where B is a subset or related. Let's check:
- B = {2,4,6,8,10}
- C = {12,14,18,20,24} → no common elements → B is not a subset of C
- A = {3,5,7,9} → no overlap
- D = {21,26,31,36} → no overlap
So B is not a subset of any of them.

But perhaps the blank is asking for membership? But B is a set, so likely we're checking subsets.

Wait — maybe the question wants us to write a relation like $ B \subset \text{something} $? But none of the given sets contain B as a subset.

Alternatively, perhaps it's a typo, or maybe we are to say $ B \not\subset C $, etc.

But looking at the pattern, it seems like:

(iv) $ B \subset \_\_\_ $ → But B is not a subset of any of the others.

Wait — let’s re-express:

Maybe it's asking: B is a subset of...? But none of the other sets contain elements of B.

Alternatively, perhaps the intention is to write:

(iv) $ B \subset \mathbb{E} $ (even numbers), but that's not given.

Since only sets A, B, C, D are given, and B is not a subset of any of them, the only possible answer is:

$ B \not\subset C $, $ B \not\subset D $, etc.

But since the blank is after $ B \subset $, and no such set exists, perhaps it's a mistake.

Wait — perhaps it's asking to complete the statement:

(iv) $ B \subset \_\_\_ $ → But no such set is present.

Alternatively, maybe the question meant to ask about subsets of B?

Wait — let's look again.

Actually, the format is:

(iv) $ B \in \_\_\_ $? No, it's written as:

(iv) $ B \subset \_\_\_ $ → but there is no set that contains B as a subset.

Wait — unless we consider universal set? But not provided.

Alternatively, perhaps it's a typo and they meant:

(iv) $ B \subset \mathbb{N} $, but again, not among options.

Alternatively, maybe the question expects:

(iv) $ B \subset \text{no set here} $ → but that’s not helpful.

Wait — perhaps we are to fill in which set B is a subset of, but since none of the others contain B, maybe it's none?

But let's double-check:

Is B ⊂ C? No — 2 ∉ C, etc.

Is B ⊂ D? No.

Is B ⊂ A? No.

So B is not a subset of any of the given sets.

But perhaps the blank is meant to be filled with a symbol?

Wait — maybe the question is:

(iv) $ B \subset \_\_\_ $ → and we have to say “none” or leave blank?

Alternatively, perhaps it's asking for which set contains B?

But no such set exists.

Wait — maybe it's a misprint, and it's supposed to be:

(iv) $ B \subset \text{Set of even numbers} $, but not given.

Alternatively, perhaps it's asking for membership of B?

No — B is a set, so $ B \in $ would require a set of sets.

But none of the sets contain B.

So likely, this part is flawed, or we interpret differently.

Wait — let’s go back to the original worksheet.

Looking at the structure:

It says:

(b) Fill in the blanks.

(i) $ 3 \in \_\_\_ $ → we fill: A

(ii) $ 4 \_\_\_ B $ → we fill: $ \in $

(iii) $ 26 \_\_\_ C $ → $ \notin $

(iv) $ B \subset \_\_\_ $ → Hmm...

Wait — maybe it's not $ B \subset \_\_\_ $, but rather $ B \subset \text{?} $

But none of the sets contain B.

Wait — unless we consider that C has even numbers, but B has smaller evens.

Still, B is not a subset of C.

Wait — let's check:

B = {2,4,6,8,10}, C = {12,14,18,20,24}

No common elements → B ∩ C = ∅ → so B is not a subset of C.

Similarly, not of D.

So B is not a subset of any of the given sets.

But maybe the question meant:

(iv) $ B \subset \mathbb{Z} $, but again, not among options.

Alternatively, perhaps it's a typo and should be:

(iv) $ B \subset \text{Set of even numbers} $, but not listed.

Wait — maybe the blank is for relation, not a set?

But the format is:

(iv) $ B \subset \_\_\_ $

So we need to fill in a set name.

But no such set exists.

Unless — wait! Is there a possibility that C contains elements from B? No.

Wait — maybe the question is asking for which set is B a subset of?

But none.

Alternatively, perhaps it's asking:

(iv) $ B \subset \_\_\_ $ → and we write “none”

But that’s not standard.

Wait — let’s look at the rest:

(v) $ 5 \_\_\_ A $ → 5 ∈ A → so $ 5 \in A $

(vi) $ 6 \_\_\_ C $ → 6 ∉ C → $ 6 \notin C $

(vii) $ 21 \_\_\_ D $ → 21 ∈ D → $ 21 \in D $

(viii) $ 16 \_\_\_ B $ → 16 ∉ B → $ 16 \notin B $

Now back to (iv): $ B \subset \_\_\_ $

Wait — perhaps it's a typo, and it should be:

(iv) $ B \subset \text{Set of even numbers} $, but not given.

Alternatively, maybe it's:

(iv) $ B \subset \mathbb{N} $ — natural numbers — but again, not among options.

Alternatively, perhaps it's meant to be:

(iv) $ B \subset \_\_\_ $ → and we are to say “none”, but that’s unlikely.

Wait — could it be that C is a set of even numbers, and B is also a set of even numbers?

But B is not a subset of C because 2 ∉ C, etc.

So no.

Alternatively, maybe the question is asking: Is B a subset of C? But the blank is after $ B \subset $, so it's expecting a set.

Wait — perhaps the blank is for a symbol?

No — the format is:

(iv) $ B \subset \_\_\_ $

So likely, it's expecting a set name, like A, B, C, D.

But none of them contain B.

So this part is problematic.

Wait — unless we consider that B is a subset of itself?

Yes! Every set is a subset of itself.

So $ B \subset B $

So (iv) $ B \subset B $

That makes sense!

Similarly, $ A \subset A $, etc.

So likely, the blank is to be filled with B

So:

(iv) $ B \subset B $

✔ That works.

Similarly, for others:

(v) $ 5 \_\_\_ A $ → $ 5 \in A $

(vi) $ 6 \_\_\_ C $ → $ 6 \notin C $

(vii) $ 21 \_\_\_ D $ → $ 21 \in D $

(viii) $ 16 \_\_\_ B $ → $ 16 \notin B $

Now let's compile:

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✔ Final Answers:

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Q1: Which are sets?

(a) Yes – well-defined: Tue, Thu
(b) No – "difficult" is subjective
(c) Yes – specific group
(d) Yes – definable rivers
(e) Yes – assuming "active" is defined
(f) Yes – mathematical definition
(g) No – "beautiful" is subjective

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Q2:

#### (a) True or False:

(i) $ 13 \in C $ → False
(ii) $ 4 \in A $ → False
(iii) $ 9 \in A $ → True
(iv) $ 24 \in C $ → True
(v) $ 5 \in D $ → False
(vi) $ 36 \in D $ → True
(vii) $ 20 \in C $ → True
(viii) $ 9 \in A $ → True

#### (b) Fill in the blanks:

(i) $ 3 \in \boxed{A} $
(ii) $ 4 \boxed{\in} B $
(iii) $ 26 \boxed{\notin} C $
(iv) $ B \subset \boxed{B} $ (since every set is a subset of itself)
(v) $ 5 \boxed{\in} A $
(vi) $ 6 \boxed{\notin} C $
(vii) $ 21 \boxed{\in} D $
(viii) $ 16 \boxed{\notin} B $

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✔ Final Answer Summary:

> Q1:
> Sets: (a), (c), (d), (e), (f)
> Not sets: (b), (g)

> Q2(a):
> (i) F (ii) F (iii) T (iv) T (v) F (vi) T (vii) T (viii) T

> Q2(b):
> (i) A
> (ii) ∈
> (iii) ∉
> (iv) B
> (v) ∈
> (vi) ∉
> (vii) ∈
> (viii) ∉

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