Certainly! Let's evaluate the statements one by one:
### Statement (i)
[tex]\[ \{b, c\} \subset \{c, d, e\} \][/tex]
A set [tex]\( A \)[/tex] is a subset of set [tex]\( B \)[/tex] (written as [tex]\( A \subset B \)[/tex]) if every element of [tex]\( A \)[/tex] is also an element of [tex]\( B \)[/tex]. Here, we can see that "b" is not an element of [tex]\(\{c, d, e\}\)[/tex]. Thus:
[tex]\[ \{b, c\} \not\subset \{c, d, e\} \][/tex]
Statement (i) is False.
### Statement (ii)
[tex]\[ \{a, e, i\} \subset \{x : x \text{ is a vowel in the English alphabet}\} \][/tex]
The set of vowels in the English alphabet is [tex]\(\{a, e, i, o, u\}\)[/tex]. Since [tex]\(\{a, e, i\}\)[/tex] is a subset of this set, the statement is true:
[tex]\[ \{a, e, i\} \subset \{a, e, i, o, u\} \][/tex]
Statement (ii) is True.
### Statement (iii)
[tex]\[ \{1, 2, 3\} \subset \{1, 2, 4, 5\} \][/tex]
For [tex]\(\{1, 2, 3\}\)[/tex] to be a subset of [tex]\(\{1, 2, 4, 5\}\)[/tex], every element in [tex]\(\{1, 2, 3\}\)[/tex] must be in [tex]\(\{1, 2, 4, 5\}\)[/tex]. However, 3 is not in [tex]\(\{1, 2, 4, 5\}\)[/tex], thus:
[tex]\[ \{1, 2, 3\} \not\subset \{1, 2, 4, 5\} \][/tex]
Statement (iii) is False.
### Statement (iv)
[tex]\[ \{c\} \subset \{b, c, d\} \][/tex]
Since set [tex]\(\{c\}\)[/tex] is entirely contained within [tex]\(\{b, c, d\}\)[/tex], the statement is true:
[tex]\[ \{c\} \subset \{b, c, d\} \][/tex]
Statement (iv) is True.
### Statement (v)
[tex]\[ \{b\} \in \{a, b, c, d\} \][/tex]
The expression [tex]\(\{b\} \in \{a, b, c, d\}\)[/tex] means checking if the set [tex]\(\{b\}\)[/tex] itself is an element of the set [tex]\(\{a, b, c, d\}\)[/tex]. However, [tex]\(\{a, b, c, d\}\)[/tex] contains elements [tex]\(a, b, c, d\)[/tex] as individual elements, not as sets. Therefore:
[tex]\[ \{b\} \notin \{a, b, c, d\} \][/tex]
Statement (v) is False.
### Statement (vi)
[tex]\[ \{y : y \text{ is an even natural number less than 6}\} \subset \{y : y \text{ is a natural number which can divide } 36\} \][/tex]
The set of even natural numbers less than 6 is [tex]\(\{2, 4\}\)[/tex]. The natural numbers that divide 36 are [tex]\(\{1, 2, 3, 4, 6, 9, 12, 18, 36\}\)[/tex]. Since [tex]\(\{2, 4\}\)[/tex] is a subset of the divisors of 36:
[tex]\[ \{2, 4\} \subset \{1, 2, 3, 4, 6, 9, 12, 18, 36\} \][/tex]
Statement (vi) is True.
So, the final answers are:
[tex]\[
\text{(i) False, (ii) True, (iii) False, (iv) True, (v) False, (vi) True}
\][/tex]
