Operations on Sets<\/h3>\n
Here we will discuss each of the sets operations in detail along with the examples.<\/p>\n
1. Union of Sets<\/strong><\/p>\n The union of two sets A and B is a set of elements that are in both A and. It is denoted by A U B.<\/p>\n So, A U B = { x | x \u2208 A or x \u2208 B }.<\/p>\n Examples:<\/p>\n If A = {1, 4, 8, 16, 32}, B = {3, 9, 27, 1, 6}<\/p>\n Now write every element of A and B in A U B without repetition.<\/p>\n A U B = {1, 3, 4, 6, 8, 9, 16, 27, 32}<\/p>\n 2. Intersection of Sets<\/strong><\/p>\n The intersection of two sets A and B means the set of elements that are common in both A and B. It is denoted by A \u2229 B.<\/p>\n Therefore, A \u2229 B = { x : x \u2208 A and x \u2208 B }.<\/p>\n Examples:<\/p>\n If A = {1, 4, 5, 10, 15, 8, 9}, B = {5, 10, 20, 25, 30}<\/p>\n Now we write the common elements from both sets A and B.<\/p>\n A \u2229 B = {5, 10}<\/p>\n 3. Difference or Complement of Set<\/strong><\/p>\n when A and B are two sets, then their difference A – B means the elements of A but not the elements of B.<\/p>\n A minus B can be written as A – B.<\/p>\n A – B = {x : x \u2208 A, and x \u2209 B}<\/p>\n A – B never equal to B – A. i.e A – B \u2260 B – A<\/p>\n If A and B are disjoint sets, then A – B = A and B – A = B.<\/p>\n Examples:<\/p>\n If A = {2, 4, 3, 6, 4, 8, 5, 10} and B = {1, 2, 3, 4}<\/p>\n Then, A – B includes elements of A but not elements of B.<\/p>\n A – B = {5, 6, 8, 10}<\/p>\n Complement of a Set<\/strong><\/p>\n The complement of a set is the set of elements that are not in that set. The complement of set a is denoted by A’.<\/p>\n Therefore, A’ = {x | x \u2209 A }<\/p>\n A’ = (U – A) here U is the universal set that contains all elements.<\/p>\n Examples:<\/p>\n A = { x : x belongs to set of even integers }<\/p>\n U = {x : x belongs to set of integers}<\/p>\n A’ = U – A<\/p>\n So, A’ = {x : x belongs to set of odd integers}<\/p>\n 4. Cartesian Product or Cross Product<\/strong><\/p>\n The cartesian product of two non-empty sets A and B are denoted by A x B. The cross product is the set of all ordered pair of elements from A and B. The cartesian product is also known as the cross product.<\/p>\n A x B = { (a, b) | a \u2208 A and b \u2208 B }<\/p>\n The cross product of two sets A x B and B x A do not contain exactly the same ordered pairs.<\/p>\n So, A x B \u2260 B x A.<\/p>\n Examples:<\/p>\n If A = {4, 5, 6} and B = (a, b}<\/p>\n The cross product of A and B have 6 ordered pairs.<\/p>\n A x B = {(4, a), (4, b), (5, a), (5, b), (6, a), (6, b)}<\/p>\n B x A = {(a, 4), (a, 5), (a, 6), (b, 4), (b, 5), (b, 6)}<\/p>\n Properties of Cartesian Product<\/strong><\/p>\n Question 1:<\/strong><\/p>\n If A = {1, 3, 5, 7}, B = {2, 4, 6, 8}, C = {1, 2, 3, 4}, D = {5, 6, 7, 8}, U = {1, 2, 3, 4, 5, 6, 7, 8} find<\/p>\n (i) A U B<\/p>\n (ii) A \u2229 C<\/p>\n (iii) D’<\/p>\n Solution:<\/strong><\/p>\n Given sets are<\/p>\n A = {1, 3, 5, 7}, B = {2, 4, 6, 8}, C = {1, 2, 3, 4}, D = {5, 6, 7, 8}, U = {1, 2, 3, 4, 5, 6, 7, 8}<\/p>\n (i) A U B = {1, 3, 5, 7} U {2, 4, 6, 8}<\/p>\n = {1, 2, 3, 4, 5, 6, 7, 8}<\/p>\n So, A U B = {1, 2, 3, 4, 5, 6, 7, 8}<\/p>\n (ii) A \u2229 C = {1, 3, 5, 7} \u2229 {1, 2, 3, 4}<\/p>\n = {1, 3}<\/p>\n So, A \u2229 C = {1, 3}<\/p>\n (iii) D’ = U – D<\/p>\n = {1, 2, 3, 4, 5, 6, 7, 8} – {5, 6, 7, 8}<\/p>\n = {1, 2, 3, 4}<\/p>\n So, D’ = {1, 2, 3, 4}<\/p>\n Question 2:<\/strong><\/p>\n If A = {10, 12, 15, 18}, B = {11, 15, 14, 16}, C = {15, 16, 18, 7} find<\/p>\n (i) A – B<\/p>\n (ii) B – A<\/p>\n (iii) A – C<\/p>\n Solution:<\/strong><\/p>\n Given sets are<\/p>\n A = {10, 12, 15, 18}, B = {11, 15, 14, 16}, C = {15, 16, 18, 7}<\/p>\n (i) A – B = {10, 12, 15, 18} – {11, 15, 14, 16}<\/p>\n = {10, 12, 18}<\/p>\n So, A – B = {10, 12, 18}<\/p>\n (ii) B – A = {11, 15, 14, 16} – {10, 12, 15, 18}<\/p>\n = {11, 14, 16}<\/p>\n So, B – A = {11, 14, 16}<\/p>\n (iii) A – C = {10, 12, 15, 18} – {15, 16, 18, 7}<\/p>\n = {10, 12}<\/p>\n So, A – C = {10, 12}<\/p>\n Question 3:<\/strong><\/p>\n If P = {a, b, d}, Q = {m, n, o}, R = {l, e, t, t, e, r} find<\/p>\n (i) P x Q<\/p>\n (ii) P x R<\/p>\n (iii) Q x R<\/p>\n Solution:<\/strong><\/p>\n Given sets are<\/p>\n P = {a, b, d}, Q = {m, n, o}, R = {l, e, t, t, e, r}<\/p>\n (i) P x Q = {a, b, d} x {m, n, o}<\/p>\n = {(a, m), (a, n), (a, o), (b, m), (b, n), (b, o), (c, m), (c, n), (c, o)}<\/p>\n So, P x Q = {(a, m), (a, n), (a, o), (b, m), (b, n), (b, o), (c, m), (c, n), (c, o)}<\/p>\n (ii) P x R = {a, b, d} x {l, e, t, t, e, r}<\/p>\n = { (a, l), (a, e), (a, t), (a, r), (b, l), (b, e), (b, t), (b, r), (d, l), (d, e), (d, t), (d, r) }<\/p>\n So, P x R = { (a, l), (a, e), (a, t), (a, r), (b, l), (b, e), (b, t), (b, r), (d, l), (d, e), (d, t), (d, r) }<\/p>\n (iii) Q x R = {m, n, o} x {l, e, t, t, e, r}<\/p>\n = {(m l), (m, e), (m, t), (m, r), (n, l), (n, e), (n, t), (n, r), (o, l), (o, e), (o, t), (o, r)}<\/p>\n So, Q x R = {(m l), (m, e), (m, t), (m, r), (n, l), (n, e), (n, t), (n, r), (o, l), (o, e), (o, t), (o, r)}<\/p>\n Read More Related Articles:<\/strong><\/p>\n![\"Operations](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/04\/operations-on-sets-1.png\")
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Set Operations and Examples<\/h3>\n