Worksheet of Sets | PDF | Set (Mathematics) | Logic - Free Printable

It looks like you've uploaded a document explaining basic concepts of sets in mathematics. However, there is no specific problem or question included in the image — it's more of an educational summary covering:

1. Sets and Members of Sets
2. Expressing Sets
3. Cardinality of Sets

Since no explicit task or problem was given (e.g., "Find the cardinality of set G", "List all elements of set D", etc.), I'll summarize what the document teaches and provide examples of possible problems that might follow from this material, along with their solutions.

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🔍 Summary of Key Concepts

#### I. Sets and Members of Set
- A set is a collection of distinct objects.
- Example: $ A = \{1, 2, 3, 4, 5, 6\} $
- We use:
- $ \in $: means "is an element of"
- $ \notin $: means "is not an element of"

✔ Examples:
- $ 1 \in A $ → True
- $ 7 \notin A $ → True
- $ 9 \notin A $ → True

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#### II. Expressing Sets
There are three ways to write sets:

1. By words:
$ B = \{\text{prime numbers less than ten}\} $

2. Roster method (listing elements):
$ C = \{2, 3, 5, 7\} $

3. Set-builder notation:
$ D = \{x \mid x < 10, x \in \text{prime}\} $

> Note: All three represent the same set: primes less than 10.

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#### III. Cardinality of Sets
- The number of elements in a set is called its cardinality, denoted by $ n(A) $.

Examples:
- $ E = \{a, b, c, d, e\} $ → $ n(E) = 5 $ → finite set
- $ F = \{1, 3, 5, 7, ..., 15\} $ → Count odd numbers from 1 to 15:
$ 1, 3, 5, 7, 9, 11, 13, 15 $ → $ n(F) = 8 $ → finite
- $ G = \{3, 6, 9, 12, ...\} $ → Multiples of 3 → infinite → uncountable
- $ H = \{..., -3, -2, -1, 0, 1, 2, 3, 4, 5, 6\} $ → All integers → infinite

> So:
> - Finite sets: countable (E, F)
> - Infinite sets: uncountable (G, H)

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🧩 Possible Problems Based on This Material

Let me now create and solve a few example problems that could be based on this content.

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✔ Problem 1: Determine membership

Given $ A = \{1, 2, 3, 4, 5, 6\} $, decide whether each statement is true or false:
a) $ 4 \in A $
b) $ 8 \in A $
c) $ 3 \notin A $

Solution:
- a) $ 4 \in A $ → True (4 is in the set)
- b) $ 8 \in A $ → False (8 is not in the set)
- c) $ 3 \notin A $ → False (3 *is* in the set)

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✔ Problem 2: Express the set using roster method

Write the set of even numbers less than 10.

Solution:
Even numbers less than 10: $ 2, 4, 6, 8 $
So, $ \{2, 4, 6, 8\} $

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✔ Problem 3: Use set-builder notation

Write the set of prime numbers greater than 5 and less than 20.

Solution:
Primes between 5 and 20: $ 7, 11, 13, 17, 19 $
Set-builder form:
$ \{x \mid 5 < x < 20, x \in \text{prime}\} $

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✔ Problem 4: Find cardinality

Let $ P = \{x \mid x \text{ is a letter in the word "MATHEMATICS"}\} $. Find $ n(P) $.

Solution:
First, list unique letters in "MATHEMATICS":
M, A, T, H, E, I, C, S → Letters: M, A, T, H, E, I, C, S
But note: repeated letters don't count twice.

So, unique letters: M, A, T, H, E, I, C, S → total of 8 distinct letters.

Thus, $ n(P) = 8 $

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✔ Problem 5: Identify finite or infinite

Is the set $ Q = \{x \mid x \in \mathbb{Z}, x > 0\} $ finite or infinite?

Solution:
This is the set of all positive integers: $ \{1, 2, 3, 4, ...\} $ → goes on forever.

So, $ Q $ is an infinite set (uncountable).

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🎯 Final Notes

If you have a specific problem or question related to this material (like finding a set’s cardinality, expressing a set, or determining membership), please upload it clearly or type it out, and I’ll solve it step-by-step.

For now, I’ve provided:
- A breakdown of the concepts
- Clear explanations
- Sample problems with solutions

Let me know if you'd like practice exercises or want to go deeper into any topic!