Let's break down each set and list its elements in tabular form.
(a) [tex]\( A = \{ x : x \in \mathbb{W}, x \leq 5 \} \)[/tex]
- [tex]\( \mathbb{W} \)[/tex] denotes the set of whole numbers (0, 1, 2, 3, ...).
- The set [tex]\( A \)[/tex] consists of whole numbers [tex]\( x \)[/tex] such that [tex]\( x \leq 5 \)[/tex].
- Therefore, [tex]\( A = \{0, 1, 2, 3, 4, 5\} \)[/tex].
(b) [tex]\( B = \{ x : x \in \mathbb{W}, -3 < x < 3 \} \)[/tex]
- [tex]\( \mathbb{W} \)[/tex] denotes the set of whole numbers.
- The set [tex]\( B \)[/tex] consists of whole numbers [tex]\( x \)[/tex] such that [tex]\( -3 < x < 3 \)[/tex].
- Therefore, [tex]\( B = \{-2, -1, 0, 1, 2\} \)[/tex].
(c) [tex]\( C = \{ x : x \text{ is divisible by } 12 \} \)[/tex]
- The set [tex]\( C \)[/tex] includes numbers that are multiples of 12.
- Therefore, [tex]\( C = \{12, 24, 36, 48, 60, 72, 84, 96, 108, 120\} \)[/tex].
(d) [tex]\( D = \{ x : x = 3p, p \in \mathbb{W}, p \leq 3 \} \)[/tex]
- The set [tex]\( D \)[/tex] consists of values [tex]\( x \)[/tex] such that [tex]\( x = 3p \)[/tex] where [tex]\( p \)[/tex] is a whole number and [tex]\( p \leq 3 \)[/tex].
- Therefore, [tex]\( D = \{0, 3, 6, 9\} \)[/tex].
(e) [tex]\( E = \left\{ x : x = a^2, a \in \mathbb{N}, 3 < a < 7 \right\} \)[/tex]
- [tex]\( \mathbb{N} \)[/tex] denotes the set of natural numbers (1, 2, 3, ...).
- The set [tex]\( E \)[/tex] consists of squares of natural numbers [tex]\( a \)[/tex] such that [tex]\( 3 < a < 7 \)[/tex].
- Therefore, [tex]\( E = \{16, 25, 36\} \)[/tex].
(f) [tex]\( F = \{ x : x = \frac{n}{n+1}, n \in \mathbb{N} \text{ and } n \leq 4 \} \)[/tex]
- The set [tex]\( F \)[/tex] consists of values [tex]\( x \)[/tex] calculated as [tex]\( x = \frac{n}{n+1} \)[/tex] where [tex]\( n \)[/tex] is a natural number and [tex]\( n \leq 4 \)[/tex].
- Therefore, [tex]\( F = \{0.5, 0.6666666666666666, 0.75, 0.8\} \)[/tex].
(g) [tex]\( G = \{ x : x \in \mathbb{N}, 3x - 2 < 5 \} \)[/tex]
- The set [tex]\( G \)[/tex] consists of natural numbers [tex]\( x \)[/tex] such that [tex]\( 3x - 2 < 5 \)[/tex].
- Therefore, [tex]\( G = \{1, 2\} \)[/tex].
(h) [tex]\( J = \left\{ x : x \in \mathbb{N}, x^2 < 16 \right\} \)[/tex]
- The set [tex]\( J \)[/tex] consists of natural numbers [tex]\( x \)[/tex] such that [tex]\( x^2 < 16 \)[/tex].
- Therefore, [tex]\( J = \{1, 2, 3\} \)[/tex].
(i) [tex]\( K = \{ x : x \text{ is a prime number which is a divisor of 42} \} \)[/tex]
- The set [tex]\( K \)[/tex] consists of prime numbers that are divisors of 42.
- Therefore, [tex]\( K = \{2, 3, 7\} \)[/tex].
(j) [tex]\( H = \{ x : x \text{ is a 2-digit natural number such that the sum of its digits is } 10 \} \)[/tex]
- The set [tex]\( H \)[/tex] consists of two-digit natural numbers where the sum of the digits is 10. For example, [tex]\( 19, 28, 37, 46, 55, 64, 73, 82, 91 \)[/tex].
- Therefore, [tex]\( H = \{19, 28, 37, 46, 55, 64, 73, 82, 91\} \)[/tex].
