Sets Worksheet 1 | PDF | Element (Mathematics) | Numbers - Free Printable

Let's solve each of the problems step by step, explaining the concepts and reasoning involved.

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Problem 1:

Given:
Universal set $ U = \{1, 3, 5, 7, 9, 11, 13\} $

We are to determine which of the following sets are subsets of $ U $. A subset means all elements of the given set must be in $ U $.

#### Sets:
- $ B = \{2, 4\} $
- $ A = \{0\} $
- $ C = \{1, 9, 5, 13\} $
- $ D = \{5, 11, 1\} $
- $ E = \{13, 7, 9, 11, 5, 3, 1\} $
- $ F = \{2, 3, 4, 5\} $

Now check each:

- B: Elements 2 and 4 → Not in $ U $ → ✘ Not a subset
- A: Element 0 → Not in $ U $ → ✘ Not a subset
- C: Elements {1, 9, 5, 13} → All in $ U $ → ✔ Subset
- D: {5, 11, 1} → All in $ U $ → ✔ Subset
- E: {13, 7, 9, 11, 5, 3, 1} → All in $ U $ → ✔ Subset (Note: order doesn't matter; duplicates removed)
- F: {2, 3, 4, 5} → 2 and 4 not in $ U $ → ✘ Not a subset

✔ Answer: Subsets of $ U $ are: C, D, E

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Problem 2:

Given:
- $ A = \{2, 3, 4, 5, 6, 7\} $
- $ B = \{2, 4, 7, 8\} $
- $ C = \{2, 4\} $

Fill in blanks with $ \subset $ or $ \not\subset $:

(a) $ B \_\_ A $
B has element 8, which is not in A → So $ B \not\subset A $

(b) $ C \_\_ A $
C = {2,4}, both in A → $ C \subset A $

(c) $ B \_\_ C $
B has 7,8; 7 and 8 not in C → $ B \not\subset C $

(d) $ \emptyset \_\_ B $
Empty set is a subset of every set → $ \emptyset \subset B $

(e) $ C \_\_ C $
Any set is a subset of itself → $ C \subset C $

(f) $ C \_\_ B $
C = {2,4}, both in B → $ C \subset B $

✔ Answers:
(a) $ \not\subset $
(b) $ \subset $
(c) $ \not\subset $
(d) $ \subset $
(e) $ \subset $
(f) $ \subset $

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Problem 3:

Which of the following is a universal set for the other four?

Options:
(a) Even natural numbers
(b) Odd natural numbers
(c) Natural numbers
(d) Negative numbers
(e) Integers

A universal set contains all the elements of the other sets.

Let’s analyze:

- (a) Even natural numbers: only even positives → does not include odds, negatives, etc.
- (b) Odd natural numbers: only odd positives → too narrow
- (c) Natural numbers: positive integers starting from 1 → includes evens and odds, but not negative numbers
- (d) Negative numbers: only negatives → doesn’t include positives
- (e) Integers: includes all positive, negative, and zero → contains all others

But wait: do the other sets (like natural numbers) fit inside integers? Yes!

- Even natural numbers ⊆ Integers
- Odd natural numbers ⊆ Integers
- Natural numbers ⊆ Integers
- Negative numbers ⊆ Integers

So Integers contain all the others.

✔ Answer: (e) The set of integers

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Problem 4: Write all subsets for the following.

A subset of a set includes all possible combinations of its elements, including empty set and the set itself.

Let’s go one by one:

(a) $ \{3\} $
Subsets: $ \{\}, \{3\} $

(b) $ \{6, 11\} $
Subsets: $ \{\}, \{6\}, \{11\}, \{6, 11\} $

(c) $ \{2, 5, 9\} $
All combinations:
- $ \{\} $
- $ \{2\}, \{5\}, \{9\} $
- $ \{2,5\}, \{2,9\}, \{5,9\} $
- $ \{2,5,9\} $

Total: 8 subsets

(d) $ \{1, 2, 6, 7\} $
This has 4 elements → $ 2^4 = 16 $ subsets
List them:
- $ \{\} $
- $ \{1\}, \{2\}, \{6\}, \{7\} $
- $ \{1,2\}, \{1,6\}, \{1,7\}, \{2,6\}, \{2,7\}, \{6,7\} $
- $ \{1,2,6\}, \{1,2,7\}, \{1,6,7\}, \{2,6,7\} $
- $ \{1,2,6,7\} $

(e) $ \{a, b, c\} $
Same as (c): 8 subsets
- $ \{\}, \{a\}, \{b\}, \{c\}, \{a,b\}, \{a,c\}, \{b,c\}, \{a,b,c\} $

(f) $ \emptyset $
Only one subset: $ \{\} $

(g) $ \{p, q, r, s\} $
4 elements → $ 2^4 = 16 $ subsets
List:
- $ \{\} $
- Singletons: $ \{p\}, \{q\}, \{r\}, \{s\} $
- Pairs: $ \{p,q\}, \{p,r\}, \{p,s\}, \{q,r\}, \{q,s\}, \{r,s\} $
- Triples: $ \{p,q,r\}, \{p,q,s\}, \{p,r,s\}, \{q,r,s\} $
- Full set: $ \{p,q,r,s\} $

✔ Summary:
Each subset list is complete based on powerset.

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Problem 5: Write all possible proper subsets

A proper subset is a subset that is not equal to the original set. So exclude the set itself.

(a) $ \{a, b, c, d\} $
Has 4 elements → total subsets: $ 2^4 = 16 $
Proper subsets: $ 16 - 1 = 15 $ (exclude the full set)

List all except $ \{a,b,c,d\} $

- $ \{\} $
- $ \{a\}, \{b\}, \{c\}, \{d\} $
- $ \{a,b\}, \{a,c\}, \{a,d\}, \{b,c\}, \{b,d\}, \{c,d\} $
- $ \{a,b,c\}, \{a,b,d\}, \{a,c,d\}, \{b,c,d\} $

(b) $ \{1,2,3\} $
Proper subsets: all except $ \{1,2,3\} $

- $ \{\} $
- $ \{1\}, \{2\}, \{3\} $
- $ \{1,2\}, \{1,3\}, \{2,3\} $

(c) $ \{p, q, r\} $
Same as above: 7 proper subsets

- $ \{\} $
- $ \{p\}, \{q\}, \{r\} $
- $ \{p,q\}, \{p,r\}, \{q,r\} $

(d) $ \{5, 10\} $
Proper subsets: exclude $ \{5,10\} $

- $ \{\} $
- $ \{5\}, \{10\} $

(e) $ \{x\} $
Proper subsets: exclude $ \{x\} $

- $ \{\} $

(f) $ \emptyset $
No proper subsets — because the only subset is $ \emptyset $, and it's not *proper* (since it equals the set).
So proper subsets: none

✔ Answer: (f) No proper subsets

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Problem 6: Find the number of subsets for a set

Number of subsets of a set with $ n $ elements is $ 2^n $

(a) Containing 3 elements → $ 2^3 = 8 $ subsets
(b) Cardinal number is 5 → $ 2^5 = 32 $ subsets

✔ Answers:
(a) 8
(b) 32

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Problem 7: Number of proper subsets

Proper subsets = $ 2^n - 1 $ (excluding the set itself)

(a) Containing 6 elements → $ 2^6 - 1 = 64 - 1 = 63 $
(b) Cardinal number is 4 → $ 2^4 - 1 = 16 - 1 = 15 $

✔ Answers:
(a) 63
(b) 15

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## ✔ Final Answers Summary:

1. Subsets of $ U = \{1,3,5,7,9,11,13\} $:

→ C, D, E

2. Fill in $ \subset $ or $ \not\subset $:

(a) $ \not\subset $
(b) $ \subset $
(c) $ \not\subset $
(d) $ \subset $
(e) $ \subset $
(f) $ \subset $

3. Universal set:

→ (e) The set of integers

4. All subsets:

(a) $ \{\}, \{3\} $
(b) $ \{\}, \{6\}, \{11\}, \{6,11\} $
(c) $ \{\}, \{2\}, \{5\}, \{9\}, \{2,5\}, \{2,9\}, \{5,9\}, \{2,5,9\} $
(d) All 16 subsets of $ \{1,2,6,7\} $
(e) $ \{\}, \{a\}, \{b\}, \{c\}, \{a,b\}, \{a,c\}, \{b,c\}, \{a,b,c\} $
(f) $ \{\} $
(g) All 16 subsets of $ \{p,q,r,s\} $

5. Proper subsets:

(a) All subsets except $ \{a,b,c,d\} $ → 15
(b) All except $ \{1,2,3\} $ → 7
(c) All except $ \{p,q,r\} $ → 7
(d) $ \{\}, \{5\}, \{10\} $ → 3
(e) $ \{\} $ → 1
(f) None → 0

6. Number of subsets:

(a) 8
(b) 32

7. Number of proper subsets:

(a) 63
(b) 15

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