Assessment - I

1. Let [tex]$E=\{x: x \text{ is an even number}, x\ \textless \ 10\}$[/tex] and [tex]$F=\{y: y \text{ is a factor of 10}\}$[/tex] be two given sets. Answer the following questions:

a) List the members of the sets [tex]$E$[/tex] and [tex]$F$[/tex].

b) Are the sets [tex]$E$[/tex] and [tex]$F$[/tex] equal or equivalent? Give reasons.

c) Are the sets [tex]$E$[/tex] and [tex]$F$[/tex] disjoint or overlapping? Give reasons.

d) Show the sets [tex]$E$[/tex] and [tex]$F$[/tex] in a Venn diagram.



Answer :

Certainly! Let's tackle each part of this problem step-by-step.

### Part a) List the members of the sets \( E \) and \( F \)

Set \( E \):
- \( E = \{ x : x \text{ is an even number, } x < 10 \} \)
- The even numbers less than 10 are: 0, 2, 4, 6, and 8.
- Therefore, \( E = \{0, 2, 4, 6, 8\} \).

Set \( F \):
- \( F = \{ y : y \text{ is a factor of 10} \} \)
- The factors of 10 are: 1, 2, 5, 10.
- Therefore, \( F = \{1, 2, 5, 10\} \).

### Part b) Are the sets \( E \) and \( F \) equal or equivalent? Give reason.

Equal Sets:
- Two sets are equal if they contain exactly the same elements.
- \( E = \{0, 2, 4, 6, 8\} \)
- \( F = \{1, 2, 5, 10\} \)
- Since the elements of \( E \) and \( F \) are not all the same, \( E \) and \( F \) are not equal.

Equivalent Sets:
- Two sets are equivalent if they have the same number of elements.
- \( E \) has 5 elements: 0, 2, 4, 6, 8.
- \( F \) has 4 elements: 1, 2, 5, 10.
- Since \( E \) and \( F \) do not have the same number of elements, \( E \) and \( F \) are not equivalent.

### Part c) Are the sets \( E \) and \( F \) disjoint or overlapping sets? Give reason.

Disjoint Sets:
- Two sets are disjoint if they have no elements in common.
- Common elements of \( E \) and \( F \) are elements that intersect both sets.
- \( E = \{0, 2, 4, 6, 8\} \)
- \( F = \{1, 2, 5, 10\} \)
- Both sets share the common element 2.
- Therefore, \( E \) and \( F \) are not disjoint.

Overlapping Sets:
- Two sets are overlapping if they have at least one element in common.
- Since \( E \) and \( F \) share the common element 2, they are overlapping sets.

### Part d) Show the sets \( E \) and \( F \) in a Venn-diagram

To represent the sets \( E \) and \( F \) in a Venn diagram:

1. Intersection (\( E \cap F \)):
- The common element of \( E \) and \( F \) is 2.
- Intersection = \(\{2\}\).

2. Only \( E \):
- The elements of \( E \) that are not in \( F \): 0, 4, 6, 8.
- Only \( E \) = \(\{0, 4, 6, 8\}\).

3. Only \( F \):
- The elements of \( F \) that are not in \( E \): 1, 5, 10.
- Only \( F \) = \(\{1, 5, 10\}\).

Here is the Venn diagram representation:

```
(Only E) (Intersection) (Only F)
{0, 4, 6, 8} {2} {1, 5, 10}

+------------------+------------------+
| | |
| | |
| {0, 4, 6, 8} | {2} | {1, 5, 10}
| | |
| | |
| | |
+------------------+------------------+
```

Therefore, we've broken down the problem and provided detailed answers for each part.