Union and Intersection

The intersection and union of sets can be confusing especially when you have to understand the words “and” and “or” which are associated with these concepts.  Think of two roads that intersect in your hometown.  Now, think of a business at the intersection of these roads.

In Shelbyville there are two roads named Main and Madison.  If you were at the intersection then you would see KFC.  KFC is at the intersection of Main and Madison.  In other words, the intersection of sets is the set of elements or numbers common to both sets.  The union of sets is like a “marriage” of elements and numbers in the sets.  When a husband and wife marry (often referred to as a union), they bring all of their belongings under one household.  Belongings which either he or she owned are now in their union (marriage.)  Every element that is in either set is in the union of sets.

The symbol for Union is  and the symbol for Intersection is .  Observe the following example and note that the union of sets is the set of elements that is in the first set “or” the second set while the intersection of sets is the set of elements that are in the first set “and” the second set.

(7)  Given that  A={2,4,6,8,10}  and  B={1,2,3,4,5}

       A B={1,2,3,4,5,6,8,10} … the Union

       A B={2,4}                     … the Intersection

The last concept in this chapter is that of absolute value equations and inequalities.  You must remember that the value of the “inside” of the absolute value symbol may be positive or negative, but once the absolute value is evaluated the result is always positive.  In other words,  or .

(8)  Solve the equation: , then  or  .  Solving these equations we would find that  or .  The absolute value equation has two points that satisfy the equation.  Checking this answer would provide the following information:                       and .
The solution to the equation is the set {-1,4}.

Absolute value inequalities are solved in a similar fashion.  You must understand that the “less than” and “less than or equal to” inequalities are compound statements (the three part inequalities that were solved in the first section).

(9)  Solve the inequality: .   This would mean that the value for  would lie between the values of  5  and  –5.  Therefore, you would set up a compound inequality to solve the problem.  The problem would now look like this:        

Solving this compound inequality would provide the result that .  Which means that the value for x lies between  –1  and  4. 

(graph)                               [             ]                                                         

                                         -1               4

(interval notation)                        … notice that this is an Intersection of two sets of numbers

The “greater than” and “greater than or equal to” inequalities are disjoint statements.  This means that you must “dis”join them or take them apart to solve.  The answer to the inequality will be greater than the positive value of the number “or” less than the negative value of the number.  The answer is the union of the two inequalities.

(10)  Solve the inequality: .   This would mean that the value for  would be greater than  5  or the value for  would be less than  –5.  When you “dis”join the inequality to solve, the problem would look like this:           “or” 

Solving these inequalities would provide the result  “or” .                                          

(graph)                                       )            (      

                                                 -1            4

(interval notation)               … notice that this is a Union of two sets of numbers