Let \( U = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \), \( A = \{1, 2, 3, 4\} \), \( B = \{2, 4, 6, 8\} \) and \( C = \{3, 4, 5, 6\} \). Find:
(i) \( A' \) (ii) \( B' \) (iii) \( (A \cup C)' \) (iv) \( (A \cup B)' \) (v) \( (A')' \) (vi) \( (B - C)' \)
(i) \( A' = \{5, 6, 7, 8, 9\} \)
(ii) \( B' = \{1, 3, 5, 7, 9\} \)
(iii) \( (A \cup C)' = \{7, 8, 9\} \)
(iv) \( (A \cup B)' = \{5, 7, 9\} \)
(v) \( (A')' = \{1, 2, 3, 4\} \)
(vi) \( (B - C)' = \{1, 3, 4, 5, 6, 7, 9\} \)
If \( U = \{a, b, c, d, e, f, g, h\} \), find the complements of the following sets:
(i) \( A = \{a, b, c\} \) (ii) \( B = \{d, e, f, g\} \) (iii) \( C = \{a, c, e, g\} \) (iv) \( D = \{f, g, h, a\} \)
(i) \( A' = \{d, e, f, g, h\} \)
(ii) \( B' = \{a, b, c, h\} \)
(iii) \( C' = \{b, d, f, h\} \)
(iv) \( D' = \{b, c, d, e\} \)
Taking the set of natural numbers as the universal set, write down the complements of the following sets:
(i) \( \{x : x \text{ is an even natural number}\} \)
(ii) \( \{x : x \text{ is an odd natural number}\} \)
(iii) \( \{x : x \text{ is a positive multiple of } 3\} \)
(iv) \( \{x : x \text{ is a prime number}\} \)
(v) \( \{x : x \text{ is a natural number divisible by } 3 \text{ and } 5\} \)
(vi) \( \{x : x \text{ is a perfect square}\} \)
(vii) \( \{x : x \text{ is a perfect cube}\} \)
(viii) \( \{x : x + 5 = 8\} \)
(ix) \( \{x : 2x + 5 = 9\} \)
(x) \( \{x : x \ge 7\} \)
(xi) \( \{x : x \in \mathbb{N} \text{ and } 2x + 1 > 10\} \)
(i) \( \{x : x \text{ is an odd natural number}\} \)
(ii) \( \{x : x \text{ is an even natural number}\} \)
(iii) \( \{x : x \in \mathbb{N} \text{ and } x \text{ is not a multiple of } 3\} \)
(iv) \( \{x : x \text{ is a positive composite number or } x = 1\} \)
(v) \( \{x : x \text{ is a positive integer which is not divisible by } 3 \text{ or not divisible by } 5\} \)
(vi) \( \{x : x \in \mathbb{N} \text{ and } x \text{ is not a perfect square}\} \)
(vii) \( \{x : x \in \mathbb{N} \text{ and } x \text{ is not a perfect cube}\} \)
(viii) \( \{x : x \in \mathbb{N} \text{ and } x \ne 3\} \)
(ix) \( \{x : x \in \mathbb{N} \text{ and } x \ne 2\} \)
(x) \( \{x : x \in \mathbb{N} \text{ and } x < 7\} \)
(xi) \( \{x : x \in \mathbb{N} \text{ and } x \le \tfrac{9}{2}\} \)
If \( U = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \), \( A = \{2, 4, 6, 8\} \) and \( B = \{2, 3, 5, 7\} \), verify that:
(i) \( (A \cup B)' = A' \cap B' \)
(ii) \( (A \cap B)' = A' \cup B' \)
Draw appropriate Venn diagram for each of the following:
(i) \( (A \cup B)' \)
(ii) \( A' \cap B' \)
(iii) \( (A \cap B)' \)
(iv) \( A' \cup B' \)
Let \( U \) be the set of all triangles in a plane. If \( A \) is the set of all triangles with at least one angle different from \(60^{\circ}\), what is \( A' \)?
\( A' \) is the set of all equilateral triangles.
Fill in the blanks to make each of the following a true statement:
(i) \( A \cup A' = \ldots \)
(ii) \( \varphi' \cap A = \ldots \)
(iii) \( A \cap A' = \ldots \)
(iv) \( U' \cap A = \ldots \)
(i) \( U \)
(ii) \( A \)
(iii) \( \varphi \)
(iv) \( \varphi \)