Disjoint Sets

While we have discussed union, intersections, and complements of two or more sets, one often forgotten component of set theory is the case where two sets are disjoint.  Two sets A and B are called disjoint if A and B have no elements in common.  Another equivalent definition of disjoint sets would be that the intersection of the two sets actually equals the empty set.  While we can debate which definition has a nicer ring to it, the main focus here is to be able to identify when two sets are disjoint.

Examples

Examples of Disjoint Sets

  1.   {1,2,3,4,5} and {a,b,c,d,e}
  2.   {Microsoft, Google, Amazon, Facebook, Twitter} and {Norfolk Southern, Union Pacific, CSX, Kansas City Southern}
  3.   {elf, Santa, reindeer} and {menorah, dreidels, latkes}
  4.   The set of all even integers and the set of all odd integers

In all of the above examples notice that none of the two sets listed on a single line share an element.  As a result, each case provides us with an example of disjoint sets.  While this may seem like an easy task, it can be harder to check as the cardinality of our sets increase.

Examples of Sets that are not Disjoint

  1.  The set of all people with first name Aaron and the set of all people with last name Rodgers
  2.  The set of all natural numbers that are multiples of 517 and the set of all integers that are multiples of 1389
  3.  The set of all presidents of the United States and the set of all people whose middle name is Abram

Depending upon our familiarity with certain topics, these examples can range from an easy solution to a more difficult one, depending upon our familiarity with the topic.  We shall explain why the above sets are not disjoint here.

Reasons the Above Sets are not Disjoint

  1. Aaron Rodgers is a football player with the Green Bay Packers that would intersect with both sets.
  2. The number 718,113 is both a natural number and integer that is a multiple of 517 and 1389.  In fact, infinitely many more examples exist, but all of those examples are multiples of 718,113.
  3. James Abram Garfield was the 20th president of the U.S. and the only president with the middle name Abram.

Learning Activities

In order to further comprehend the material, please attempt the following exercises.

  1. Determine which of the following sets are disjoint.
  2. Create an example of two disjoint sets.
  3. Provide an example of two infinite disjoint sets.