Find the union of each of the following pairs of sets:
(i) \( X = \{1, 3, 5\},\ Y = \{1, 2, 3\} \)
(ii) \( A = \{a, e, i, o, u\},\ B = \{a, b, c\} \)
(iii) \( A = \{x : x \text{ is a natural number and multiple of }3\},\ B = \{x : x \text{ is a natural number less than }6\} \)
(iv) \( A = \{x : x \text{ is a natural number and }1 < x \le 6\},\ B = \{x : x \text{ is a natural number and }6 < x < 10\} \)
(v) \( A = \{1, 2, 3\},\ B = \varphi \)
(i) \( X \cup Y = \{1, 2, 3, 5\} \)
(ii) \( A \cup B = \{a, b, c, e, i, o, u\} \)
(iii) \( A \cup B = \{x : x = 1, 2, 4, 5 \text{ or a multiple of }3\} \)
(iv) \( A \cup B = \{x : 1 < x < 10,\ x \in \mathbb{N}\} \)
(v) \( A \cup B = \{1, 2, 3\} \)
Let \( A = \{a, b\},\ B = \{a, b, c\} \). Is \( A \subset B \)? What is \( A \cup B \)?
Yes, \( A \subset B \) and \( A \cup B = \{a, b, c\} \).
If \( A \) and \( B \) are two sets such that \( A \subset B \), then what is \( A \cup B \)?
\( A \cup B = B \).
If \( A = \{1, 2, 3, 4\},\ B = \{3, 4, 5, 6\},\ C = \{5, 6, 7, 8\} \) and \( D = \{7, 8, 9, 10\} \), find:
(i) \( A \cup B \)
(ii) \( A \cup C \)
(iii) \( B \cup C \)
(iv) \( B \cup D \)
(v) \( A \cup B \cup C \)
(vi) \( A \cup B \cup D \)
(vii) \( B \cup C \cup D \)
(i) \( \{1, 2, 3, 4, 5, 6\} \)
(ii) \( \{1, 2, 3, 4, 5, 6, 7, 8\} \)
(iii) \( \{3, 4, 5, 6, 7, 8\} \)
(iv) \( \{3, 4, 5, 6, 7, 8, 9, 10\} \)
(v) \( \{1, 2, 3, 4, 5, 6, 7, 8\} \)
(vi) \( \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \)
(vii) \( \{3, 4, 5, 6, 7, 8, 9, 10\} \)
Find the intersection of each pair of sets of Question 1 above.
(i) \( X \cap Y = \{1, 3\} \)
(ii) \( A \cap B = \{a\} \)
(iii) \( \{3\} \)
(iv) \( \varphi \)
(v) \( \varphi \)
If \( A = \{3, 5, 7, 9, 11\},\ B = \{7, 9, 11, 13\},\ C = \{11, 13, 15\} \) and \( D = \{15, 17\} \), find:
(i) \( A \cap B \)
(ii) \( B \cap C \)
(iii) \( A \cap C \cap D \)
(iv) \( A \cap C \)
(v) \( A \cap D \)
(vi) \( A \cap (B \cup D) \)
(vii) \( A \cap D \)
(viii) \( A \cap (B \cup D) \)
(ix) \( (A \cap B) \cap (B \cup C) \)
(x) \( (A \cup D) \cap (B \cup C) \)
(i) \( \{7, 9, 11\} \)
(ii) \( \{11, 13\} \)
(iii) \( \varphi \)
(iv) \( \{11\} \)
(v) \( \varphi \)
(vi) \( \{7, 9, 11\} \)
(vii) \( \varphi \)
(viii) \( \{7, 9, 11\} \)
(ix) \( \{7, 9, 11\} \)
(x) \( \{7, 9, 11, 15\} \)
If \( A = \{x : x \text{ is a natural number}\} \), \( B = \{x : x \text{ is an even natural number}\} \), \( C = \{x : x \text{ is an odd natural number}\} \) and \( D = \{x : x \text{ is a prime number}\} \), find:
(i) \( A \cap B \)
(ii) \( A \cap C \)
(iii) \( A \cap D \)
(iv) \( B \cap C \)
(v) \( B \cap D \)
(vi) \( C \cap D \)
(i) \( B \)
(ii) \( C \)
(iii) \( D \)
(iv) \( \varphi \)
(v) \( \{2\} \)
(vi) \( \{x : x \text{ is an odd prime number}\} \)
Which of the following pairs of sets are disjoint?
(i) \( \{1, 2, 3, 4\} \) and \( \{x : x \text{ is a natural number and }4 \le x \le 6\} \)
(ii) \( \{a, e, i, o, u\} \) and \( \{c, d, e, f\} \)
(iii) \( \{x : x \text{ is an even integer}\} \) and \( \{x : x \text{ is an odd integer}\} \)
Only pair (iii) is disjoint.
If \( A = \{3, 6, 9, 12, 15, 18, 21\},\ B = \{4, 8, 12, 16, 20\},\ C = \{2, 4, 6, 8, 10, 12, 14, 16\},\ D = \{5, 10, 15, 20\} \), find:
(i) \( A - B \)
(ii) \( A - C \)
(iii) \( A - D \)
(iv) \( B - A \)
(v) \( C - A \)
(vi) \( D - A \)
(vii) \( B - C \)
(viii) \( C - B \)
(ix) \( C - D \)
(x) \( D - B \)
(xi) \( C - D \)
(xii) \( D - C \)
(i) \( \{3, 6, 9, 15, 18, 21\} \)
(ii) \( \{3, 9, 15, 18, 21\} \)
(iii) \( \{3, 6, 9, 12, 18, 21\} \)
(iv) \( \{4, 8, 16, 20\} \)
(v) \( \{2, 4, 8, 10, 14, 16\} \)
(vi) \( \{5, 10, 20\} \)
(vii) \( \{20\} \)
(viii) \( \{4, 8, 12, 16\} \)
(ix) \( \{2, 6, 10, 14\} \)
(x) \( \{5, 10, 15\} \)
(xi) \( \{2, 4, 6, 8, 12, 14, 16\} \)
(xii) \( \{5, 15, 20\} \)
If \( X = \{a, b, c, d\} \) and \( Y = \{f, b, d, g\} \), find:
(i) \( X - Y \)
(ii) \( Y - X \)
(iii) \( X \cap Y \)
(i) \( \{a, c\} \)
(ii) \( \{f, g\} \)
(iii) \( \{b, d\} \)
If \( R \) is the set of real numbers and \( Q \) is the set of rational numbers, then what is \( R - Q \)?
\( R - Q \) is the set of irrational numbers.
State whether each of the following statement is true or false. Justify your answer.
(i) \( \{2, 3, 4, 5\} \) and \( \{3, 6\} \) are disjoint sets.
(ii) \( \{a, e, i, o, u\} \) and \( \{a, b, c, d\} \) are disjoint sets.
(iii) \( \{2, 6, 10, 14\} \) and \( \{3, 7, 11, 15\} \) are disjoint sets.
(iv) \( \{2, 6, 10\} \) and \( \{3, 7, 11\} \) are disjoint sets.
(i) False
(ii) False
(iii) True
(iv) True