More about Sets
What is a set? A set is a well defined collection of objects. Is S = {some numbers} a set? Is W = {Fred, George, Ron} a set?
What is in a set? The objects in a set are called elements or members. We write 3S or 6 {1, 2, 3, …} Also, Harry W
How do you write a set? They are enclosed by brackets and the elements are separated by commas. The series of dots (… ) are called ellipses and mean “the pattern continues”.
How do you write a set? Sets can be written by rule S = {x| 1 < x < 10 and x I} or by roster S = {2, 3, 4, 5, …,9}
More details... Two sets are equal iff they contain the same elements. If A = {2, 3, 4, … 11} and if B = {x|1 < x < 12 and x I}, is A = B?
More details... U usually stands for the universal set. We are usually given the universal set. It is often a set of numbers, such as I, E or R.
More details... The complement of a set (written A’ or Ā ) contains everything in the universal set that isn’t in the set itself. If U = {a, b, c, … f} and if A = {f, a, c, e}, find A’.
More details... Set A is a subset of B ( A B) if and only if every member of A is also in B. If B = {a, b, c, … f} and if A = {f, a, c, e}, then A B. Is A’ B ?
More details... Set A is a proper subset of B ( A B) if and only if A B but A B. A = {a, c, e}, B = {f, a, c, e} and C = {a, c, e, f}, Then A B, and A B. Furthermore, B C, but B C.
More details... A set with no elements is called the empty set and is written as {} or . The empty set is a subset of all sets.
More details... The cardinality of a set is the number of elements in a set, and is written |S|. If K = {3, 5, 8}, then |K| = 3 If P is the set of all subsets of K, find |P|.
More details... The cardinality of a set can only be found if the set is finite. But infinite sets contain an infinite number of elements.
More details... The union of A and B (written AB) is the set that contains all elements of A and all elements of B.
More details... The intersection of A and B (written AB) is the set that contains all elements that A and B have in common.
More details... If the intersection of A and B is the empty set (written AB = ), then A and B are disjoint sets.
More details... The difference set of A and B (written A - B) is the set that contains all elements in A that are not in B.
More details... The set of elements that are either in A or B, but not both is called the symmetric difference of A and B and is written A B.
Example If L = {1, 3}, M = {2, 3, 4}, N = {3, 4}, O = {2, 4, 5}, and U = {1, 2, 3, 4, 5} Is N M ? Is L M? Is N M?
Example If L = {1, 3}, M = {2, 3, 4}, N = {3, 4}, O = {2, 4, 5}, and U = {1, 2, 3, 4, 5} Write N’. Write N O. Write N O.
Example If L = {1, 3}, M = {2, 3, 4}, N = {3, 4}, O = {2, 4, 5}, and U = {1, 2, 3, 4, 5} Write |M O|. Write |O’|. Write M - O.
Example A B C D AC BD = AC BD = AB CD =
Example A B C D F CA CF = AD CF = AB CF =
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