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Assessment - I

Given sets:
[tex]\[ E = \{x \mid x \text{ is an even number, } x \ \textless \ 10\} \][/tex]
[tex]\[ F = \{y \mid y \text{ is a factor of } 10\} \][/tex]

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 a reason.

c) Are the sets [tex]\( E \)[/tex] and [tex]\( F \)[/tex] disjoint or overlapping? Give a reason.

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



Answer :

Let's solve the questions step-by-step based on the given sets [tex]\( E \)[/tex] and [tex]\( F \)[/tex].

### a) List the members of the sets [tex]\( E \)[/tex] and [tex]\( F \)[/tex].

Set [tex]\( E \)[/tex]:
[tex]\( E \)[/tex] is defined as the set of all even numbers less than 10. The even numbers less than 10 are:
[tex]\[ 0, 2, 4, 6, 8 \][/tex]

Thus, the members of set [tex]\( E \)[/tex] are:
[tex]\[ E = \{0, 2, 4, 6, 8\} \][/tex]

Set [tex]\( F \)[/tex]:
[tex]\( F \)[/tex] is defined as the set of all factors of 10. The factors of 10 are:
[tex]\[ 1, 2, 5, 10 \][/tex]

Thus, the members of set [tex]\( F \)[/tex] are:
[tex]\[ F = \{1, 2, 5, 10\} \][/tex]

### b) Are the sets [tex]\( E \)[/tex] and [tex]\( F \)[/tex] equal or equivalent? Give reason.

Equality:
Two sets are equal if they contain the same elements. Comparing the members of sets [tex]\( E \)[/tex] and [tex]\( F \)[/tex]:

[tex]\[ E = \{0, 2, 4, 6, 8\} \][/tex]
[tex]\[ F = \{1, 2, 5, 10\} \][/tex]

Since [tex]\( E \)[/tex] contains 0, 4, 6, and 8 which are not in [tex]\( F \)[/tex] and [tex]\( F \)[/tex] contains 1, 5, and 10 which are not in [tex]\( E \)[/tex], the sets do not have the same elements.

Therefore, the sets [tex]\( E \)[/tex] and [tex]\( F \)[/tex] are not equal.

Equivalence:
Two sets are equivalent if they have the same number of elements. The number of elements in each set is:

[tex]\[ |E| = 5 \][/tex]
[tex]\[ |F| = 4 \][/tex]

Since [tex]\( E \)[/tex] has 5 elements and [tex]\( F \)[/tex] has 4 elements, they do not have the same number of elements.

Therefore, the sets [tex]\( E \)[/tex] and [tex]\( F \)[/tex] are not equivalent.

### c) Are the sets [tex]\( E \)[/tex] and [tex]\( F \)[/tex] disjoint or overlapping? Give reason.

Disjoint sets:
Two sets are disjoint if they have no elements in common. Let's check for common elements between [tex]\( E \)[/tex] and [tex]\( F \)[/tex]:

[tex]\[ E = \{0, 2, 4, 6, 8\} \][/tex]
[tex]\[ F = \{1, 2, 5, 10\} \][/tex]

The common element between [tex]\( E \)[/tex] and [tex]\( F \)[/tex] is:
[tex]\[ \{2\} \][/tex]

Since there is a common element, the sets [tex]\( E \)[/tex] and [tex]\( F \)[/tex] are not disjoint but overlapping.

Therefore, the sets [tex]\( E \)[/tex] and [tex]\( F \)[/tex] are overlapping.

### d) Show the sets [tex]\( E \)[/tex] and [tex]\( F \)[/tex] in a Venn-diagram.

Since this is a textual solution and we cannot actually draw a Venn-diagram, we will describe it.

1. Draw two overlapping circles.
2. Label the left circle as [tex]\( E \)[/tex] and the right circle as [tex]\( F \)[/tex].
3. Place the common element [tex]\( \{2\} \)[/tex] in the overlapping region.
4. Place the remaining elements of [tex]\( E \)[/tex] (\{0, 4, 6, 8\}) in the non-overlapping part of the left circle.
5. Place the remaining elements of [tex]\( F \)[/tex] (\{1, 5, 10\}) in the non-overlapping part of the right circle.

The Venn-diagram should visually represent the intersection at [tex]\( \{2\} \)[/tex], and the other elements distributed in their respective sets.

This detailed solution addresses each part of the question comprehensively using the information provided.