OPERATIONS OF SETS
Example 1:
If A = {. . ., -3, -2, -1} and B = {0, 1, 2, 3, . . .}, then A ∪ B = {. . ., -3, -2, -1, 0, 1, 2, 3, . . .}, or the Integers.
Example 2:
If C = {a, b, c, d}, and D = {a, c}, then C ∩ D = {a, c}. Notice that in example 2, all the elements of D are in C. Therefore, the following property is evident:
Property:
If D C, then C ∩ D = D.
Example 2:
If P = {1, 2, 3} and Q = {2, 3, 4}, then P ∪ Q = {1, 2, 3, 4}. Notice that two and three appear in both sets, but only once each in the union set. Once you include an element in the union, it is unnecessary to include it twice.
the difference between the union and intersection of two sets.
The union of two sets is a new set that contains all of the elements that are in at least one of the two sets. The union is written as A ∪ B . The intersection of two sets is a new set that contains all of the elements that are in both sets. The intersection is written as A ∩ B .
Which of the following are operations in the algebra of sets?
multiplication union intersection
Match the following. Given A = {1, 2, 3, 4, 5} B = {2, 4, 6} C = {1, 3, 5}
A ∪ B={1, 2, 3, 4, 5, 6} A ∪ C={1, 2, 3, 4, 5} A ∩ B={2, 4} A ∩ C={1, 3, 5} B ∩ C=null, or empty set
Match the following. Given A = {1, 2, 3, 4, 5} B = {2, 4, 6} C = {1, 3, 5}
B ∪ C={1, 2, 3, 4, 5, 6} A ∩ B={2, 4} A ∩ C={1, 3, 5} A ∩ (B ∩ C)=null, or empty set A ∪ (B ∩ C)={1, 2, 3, 4, 5}
Match the following. Given D = {x | x is a whole number} E = {x | x is a perfect square < 100} F = {x | x is an even number between 10 and 20}
D ∩ E={1, 4, 9, 16, 25, 36, 49, 64, 81, 12, 14, 18} D ∩ F={12, 14, 16, 18} D ∩ (E ∩ F)={1, 4, 9, 16, 25, 36, 49, 64, 81} D ∪ (E ∩ F)={all whole numbers} D ∩ (E ∪ F)={16}
Example 1:
If A = {1, 2, 3} and B = {2, 3, 4}, then A ∩ B = {2, 3}, the elements in both set A and set B.
Construct the appropriate number sets with the given information. Use the sets to match the following: Given a = ∅ b = B c = then A ∪ B = B d = A e = then A ∩ B = B
If A ⊂ B=b If B ⊂ A=a A ∪ ∅=d A ∩ ∅=c B ∪ ∅=e
D = {x|x is a whole number} E = {x|x is a perfect square between 1 and 9} F = {x|x is an even number greater than or equal to 2 and less than 9} Which of the following is an element of D ∩ (E ∩ F)?
4
Example 3:
Now suppose E = {odd numbers} and F = {even numbers}. Since all numbers are either even or odd but not both, the intersection E ∩ F is the empty set.
D = {x|x is a whole number} E = {x|x is a perfect square between 1 and 9} F = {x|x is an even number greater than or equal to 2 and less than 9} Which of the following is D ∩ F?
{2, 4, 6, 8}
D = {x|x is a whole number} E = {x|x is a perfect square between 1 and 9} F = {x|x is an even number greater than or equal to 2 and less than 9} Which of the following is E ∪ F?
{4}
Select the statements that are true based on the following given information. D = {x | x is a whole number} E = {x | x is a perfect square < 36} F = {x | x is an even number between 20 and 30}
The expression D ∩ F is {12, 14, 16, 18}. (E ∩ F) is the empty set.
Select the correct answer. Given D = {x | x is a whole number} E = {x | x is a perfect square between 49 and 100} F = {x | x is an even number between 10 and 20}
The expression D ∪ F means . . . all whole numbers The expression D ∩ F means . . . 12, 14, 16, 18 The expression D ∩ E means . . . 64, 81 The expression E ∩ F means . . . the empty set The expression D ∩ (E ∪ F) means . . . empty set
Operations of Sets: Intersection
The intersection of two sets, A and B, is a set whose elements are those that are common to A and B. This means the set includes only the elements that are found in both Set A and Set B. The symbol for intersection is ∩.
union
The union of two sets, A and B, is a set whose elements are those that appear in either Set A or Set B without repetition. Without repetition means that elements in both A and B need to be listed only once in the union. The union combines every element from Set A and Set B. The symbol for union is ∪.
