Set Concepts

• A set is a collection of objects.
  • The collection of all presidents of the United States.
  • The collection of all even integers between 0 and 10 inclusive.
• The study of sets introduces a type of mathematics many of us have not yet seen.
  • Arithmetic
  • Algebra
  • Geometry
  • Probability and Statistics
  • These all give us a different view of the world according to mathematicians.
• The objects in a set are called elements
  • Abraham Lincoln is an element of the set of presidents of the United States.
  • 2 is an element of the set of even integers between 0 and 10 inclusive.
• A set is well defined if the items are clearly identifiable
  • The set of presidents of the United States is well defined.
  • The set of the five best presidents of the United States is not well defined.
• There are three methods for specifying sets
• The descriptive method requires us to describe the set
  • Let P be the set of presidents of the United States.
  • Let S be the set of even integers between 0 and 10 inclusive.
• Roster from requires us to list the elements of a set.
  • Let P = { George Washington, John Adams, Thomas Jefferson, James Madison, James Monroe, John Quincy Adams, Andrew Jackson, Martin Van Buren, William Henry Harrison, John Tyler, James Knox Polk, Zachary Taylor, Millard Fillmore, Franklin Pierce, James Buchanan, Abraham Lincoln, Andrew Johnson, Ulysses S. Grant, Rutherford Birchard Hayes, James Abram Garfield, Chester Alan Arthur, Grover Cleveland, Benjamin Harrison, William McKinley, Theodore Roosevelt, William Howard Taft, Woodrow Wilson, Warren Gamaliel Harding, Calvin Coolidge, Herbert Clark Hoover, Franklin Delano Roosevelt, Harry S. Truman, Dwight David Eisenhower, John Fitzgerald Kennedy, Lyndon Baines Johnson, Richard Milhous Nixon, Gerald Rudolph Ford, James Earl Carter, Ronald Wilson Reagan, George Herbert Walker Bush, William Jefferson Clinton, George Walker Bush, Barack Hussein Obama}
  • Let S = {0,2,4,6,8,10}
  • Notice in this form, we use {} to enclose the set
  • Elements are separated by a comma (,)
• If the set can be listed by a pattern, ellipses are used
  • The set of Natural Numbers N = { 1, 2, 3, 4, ...}
  • Even integers between 0 and 100 inclusive E = { 0, 2, 4, 6, ..., 100}
  • Or the set of Integers, I = { ..., -3, -2, -1, 0, 1, 2, 3, ...}
• The symbol ∈ is used to show that something is an element of a set.
  • George Walker Bush ∈ P
  • Mitt Romney ∉ P
  • 5 ∈ N
  • 0 ∉ N
• Set Builder Notation is the final way to specify a set.
  • E = { x | x ∈ I and 0 ≤ x ≤ 100}
  • "Let E be the set of x such that x is in the integers and x is greater than or equal to zero and x is less than or equal to 100"
• Each type of notation is useful and you should be able to use all of them.
• A set is finite if it is either empty, or the number of elements in the set is a natural number.
  • P is finite, as it contains 43 elements (Grover Cleveland is only counted once)
  • N is said to be infinite
• The cardinal number of a set A, symbolized by n(A), is the number of elements in a set
  • n(P) = 43
  • n(E) = 6
• Set A is equal to set B, symbolized by A=B, if and only if set A and set B contain exactly the same elements.
• Set A is equivalent to set B if n(A) = n(B)
  • A = { 1, 2, 3, 4}
  • B = { a, b, c, d}
  • C = { 2, 4, 1, 3}
  • D = { x, y, z}
  • Discuss if A,B,C,D are equal, equivalent or not related.
• The empty set is a set which contains no elements.
  • {}
  • Ø
• The universal set is the set of all elements under consideration for any specific discussion.

• Which of the following are well defined sets
  • The set of all students in my math 104 classes
  • The set of all integers greater than 10
  • The set of the ten best musicals
  • The set of the worst instructors at Edinboro.
• Which of the following sets are finite
  • The set of the characters in the musical Oklahoma
  • The set of integers
  • The set of planets orbiting our sun.
  • The set of atoms in our sun.
  • Be careful if you are asked to construct a finite or infinite set.
• Write the following set in roster format
  • Let S be the set of months with less than 30 days.
  • Let T be the positive multiples of ten less than 100
  • Let T be the positive multiples of ten less than 1000
  • Let T be the positive multiples of ten
  • Let B = { x | x ∈ N and x/3 ∈ N}
  • Let P be the set of Great Lakes that border California.
  Express in set builder notation
  • B = { 15,16,17,18,19,20,21,22}
  • C = { 4, 8, 12, 16, ...}
  • E is the set of even natural numbers.
  Give a descriptive form of the set
  • B = { 1,2,3,4,5,6,7}
  • C = {April, August}