[tex]$P = \{ \text{even numbers between 2 and 10} \}$[/tex]
[tex]$Q = \{ \text{prime numbers up to 6} \}$[/tex]

(a) Are sets [tex]$P$[/tex] and [tex]$Q$[/tex] disjoint? Give a reason.

(b) Write all the proper subsets of set [tex]$Q$[/tex].



Answer :

Certainly! Let's analyze the problem step by step.

### Step (a): Determine if sets [tex]\( P \)[/tex] and [tex]\( Q \)[/tex] are disjoint

- Set [tex]\( P \)[/tex]: [tex]\( \{2, 4, 6, 8, 10\} \)[/tex]
- These are the even numbers between 2 and 10.
- Set [tex]\( Q \)[/tex]: [tex]\( \{2, 3, 5\} \)[/tex]
- These are the prime numbers up to 6.

For two sets to be disjoint, they must have no elements in common. Let's find the intersection of sets [tex]\( P \)[/tex] and [tex]\( Q \)[/tex]:
- The common elements between [tex]\( P \)[/tex] and [tex]\( Q \)[/tex] are just {2} which means the intersection is {2}

Since {2} is not an empty set, sets [tex]\( P \)[/tex] and [tex]\( Q \)[/tex] are not disjoint. Therefore, there is at least one element common to both sets. The answer is: No, [tex]\( P \)[/tex] and [tex]\( Q \)[/tex] are not disjoint sets.

### Step (b): Proper subsets of set [tex]\( Q \)[/tex]

The set Q is [tex]\( \{2, 3, 5\} \)[/tex].

A proper subset of a set is any subset of that set which is not equal to the set itself. Proper subsets do not include the set itself.

Let's list all the proper subsets:

1. [tex]\( \emptyset \)[/tex] (The empty set)
2. [tex]\( \{2\} \)[/tex]
3. [tex]\( \{3\} \)[/tex]
4. [tex]\( \{5\} \)[/tex]
5. [tex]\( \{2, 3\} \)[/tex]
6. [tex]\( \{2, 5\} \)[/tex]
7. [tex]\( \{3, 5\} \)[/tex]

Therefore, all the proper subsets of set [tex]\( Q \)[/tex] are:
[tex]\[ \emptyset, \{2\}, \{3\}, \{5\}, \{2, 3\}, \{2, 5\}, \{3, 5\} \][/tex]

These are all the proper subsets of the set [tex]\( Q \)[/tex].

In summary:
- (a) [tex]\( P \)[/tex] and [tex]\( Q \)[/tex] are not disjoint sets because they share a common element, 2.
- (b) The proper subsets of [tex]\( Q \)[/tex] are [tex]\( \emptyset \)[/tex], [tex]\( \{2\} \)[/tex], [tex]\( \{3\} \)[/tex], [tex]\( \{5\} \)[/tex], [tex]\( \{2, 3\} \)[/tex], [tex]\( \{2, 5\} \)[/tex], and [tex]\( \{3, 5\} \)[/tex].