Sure, let's go through each part of the question step by step to understand how to arrive at the solution.
### (a) Type of the Relation Between Set [tex]\( M \)[/tex] and Set [tex]\( N \)[/tex]
First, let's identify the elements of the sets [tex]\( M \)[/tex] and [tex]\( N \)[/tex]:
- [tex]\( M = \{ \text{prime numbers less than 10} \} = \{2, 3, 5, 7\} \)[/tex]
- [tex]\( N = \{ \text{odd numbers less than 10} \} = \{1, 3, 5, 7, 9\} \)[/tex]
We need to determine the type of the relation between these sets. Specifically, we look for common elements between the sets:
- [tex]\( M \cap N = \{3, 5, 7\} \)[/tex]
There are common elements between sets [tex]\( M \)[/tex] and [tex]\( N \)[/tex]. Since the intersection of the two sets is not empty, the relation between set [tex]\( M \)[/tex] and set [tex]\( N \)[/tex] is an intersection.
Answer: The type of relation between set [tex]\( M \)[/tex] and set [tex]\( N \)[/tex] is an intersection.
### (b) Number of Improper Subsets of Set [tex]\( M \)[/tex]
An improper subset of a set is a subset that includes the set itself. In other words, a subset which is not proper because it is exactly the original set.
For set [tex]\( M \)[/tex], the only improper subset is the set [tex]\( M \)[/tex] itself.
Answer: There is 1 improper subset of set [tex]\( M \)[/tex].
### (c) Number of Proper Subsets of Set [tex]\( N \)[/tex]
A proper subset of a set is a subset that does not include the set itself, i.e., it is strictly smaller.
For a set with [tex]\( n \)[/tex] elements, the total number of subsets is [tex]\( 2^n \)[/tex]. However, one of these subsets is the set itself, which is not considered a proper subset.
First, we count the number of elements in set [tex]\( N \)[/tex]:
- [tex]\( |N| = 5 \)[/tex]
The total number of subsets of [tex]\( N \)[/tex] is [tex]\( 2^5 = 32 \)[/tex].
Since we exclude the set itself to count only the proper subsets:
- Number of proper subsets of [tex]\( N = 32 - 1 = 31 \)[/tex]
Answer: There are 31 proper subsets of set [tex]\( N \)[/tex].
Putting it all together:
(a) The relation between set [tex]\( M \)[/tex] and set [tex]\( N \)[/tex] is an intersection.
(b) The number of improper subsets of set [tex]\( M \)[/tex] is 1.
(c) The number of proper subsets of set [tex]\( N \)[/tex] is 31.
