Universal Set:  A universal set, sometimes called the universe, is the set of all items under consideration for a particular problem or situation.  We will let set U, unless otherwise defined, represent the universe in a given problem or situation.  We always need to be aware of the specific universal set used in any problem.  The universal set chosen can make a huge difference in the answer to a problem.

Example:  In the opening problem above, the universal set is the “set of the thirty students that went on the camping trip.”

Example:  Find the solution set for the equation, x + 5 = 3.
For the universal set N = the set of natural numbers, there is no natural number solution to this equation, i.e., the solution set is the empty set.
For the universal set U = the set of integers = {. . . −3, −2, −1,  0,  1,  2,  3, . . .}, −2 satisfies the equation since −2 + 5 = 3. So, the solution set is {−2}.

Note: The problem has a different solution set depending on the universe used, ∅ or {−2}.

Venn Diagrams:  A Venn diagram is a drawing that can be used to show how sets are related.  
        A rectangle is used to represent the universal set (universe). (Remember that the universe is everything under consideration for the given problem.)
        Circles are used to represent subsets of the universe, i.e., sets that exist within the universe. The relative placement of the circles shows how those sets are related.  In the last session, we worked with the concept of subsets. 
        A Venn diagram of  A ⊆ B looks like the following: