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This chapter covers principles of sets and counting. After completing this chapter students should be able to: use set theory and venn diagrams to solve counting problems; use the multiplication axiom to solve counting problems; use permutations to solve counting problems; use combinations to solve counting problems; and use the binomial theorem to expand x+y^n.

Chapter overview

In this chapter, you will learn to:

  1. Use set theory and Venn diagrams to solve counting problems.
  2. Use the Multiplication Axiom to solve counting problems.
  3. Use Permutations to solve counting problems.
  4. Use Combinations to solve counting problems.
  5. Use the Binomial Theorem to expand x + y n size 12{ left (x+y right ) rSup { size 8{n} } } {} .

Sets

In this section, we will familiarize ourselves with set operations and notations, so that we can apply these concepts to both counting and probability problems. We begin by defining some terms.

A set is a collection of objects, and its members are called the elements of the set. We name the set by using capital letters, and enclose its members in braces. Suppose we need to list the members of the chess club. We use the following set notation.

C = Ken, Bob, Tran, Shanti, Eric size 12{C= left lbrace "Ken, Bob, Tran, Shanti, Eric" right rbrace } {}

A set that has no members is called an empty set . The empty set is denoted by the symbol Ø.

Two sets are equal if they have the same elements.

A set A size 12{A} {} is a subset of a set B size 12{B} {} if every member of A size 12{A} {} is also a member of B size 12{B} {} .

Suppose C = Al, Bob, Chris, David, Ed size 12{C= left lbrace "Al, Bob, Chris, David, Ed" right rbrace } {} and A = Bob, David size 12{A= left lbrace "Bob, David" right rbrace } {} . Then A size 12{A} {} is a subset of C size 12{C} {} , written as A C size 12{A subseteq C} {} .

Every set is a subset of itself, and the empty set is a subset of every set.

Union of two sets

Let A size 12{A} {} and B size 12{B} {} be two sets, then the union of A size 12{A} {} and B size 12{B} {} , written as A B size 12{A union B} {} , is the set of all elements that are either in A size 12{A} {} or in B size 12{B} {} , or in both A size 12{A} {} and B size 12{B} {} .

Intersection of two sets

Let A size 12{A} {} and B size 12{B} {} be two sets, then the intersection of A size 12{A} {} and B size 12{B} {} , written as A B size 12{A intersection B} {} , is the set of all elements that are common to both sets A size 12{A} {} and B size 12{B} {} .

A universal set U size 12{U} {} is the set consisting of all elements under consideration.

Complement of a set

Let A size 12{A} {} be any set, then the complement of set A size 12{A} {} , written as A ˉ size 12{ { bar {A}}} {} , is the set consisting of elements in the universal set U size 12{U} {} that are not in A size 12{A} {} .

Disjoint sets

Two sets A size 12{A} {} and B size 12{B} {} are called disjoint sets if their intersection is an empty set.

List all the subsets of the set of primary colors red, yellow, blue size 12{ left lbrace "red, yellow, blue" right rbrace } {} .

The subsets are ∅, red size 12{ left lbrace "red" right rbrace } {} , yellow size 12{ left lbrace "yellow" right rbrace } {} , blue size 12{ left lbrace "blue" right rbrace } {} , red, yellow size 12{ left lbrace "red, yellow" right rbrace } {} , red, blue size 12{ left lbrace "red, blue" right rbrace } {} , yellow, blue size 12{ left lbrace "yellow, blue" right rbrace } {} , red, yellow, blue size 12{ left lbrace "red, yellow, blue" right rbrace } {}

Note that the empty set is a subset of every set, and a set is a subset of itself.

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Let F = Aikman, Jackson, Rice, Sanders, Young size 12{F= left lbrace "Aikman, Jackson, Rice, Sanders, Young" right rbrace } {} , and B = Griffey, Jackson, Sanders, Thomas size 12{B= left lbrace "Griffey, Jackson, Sanders, Thomas" right rbrace } {} . Find the intersection of the sets F size 12{F} {} and B size 12{B} {} .

The intersection of the two sets is the set whose elements belong to both sets. Therefore,

F B = Jackson, Sanders size 12{F intersection B= left lbrace "Jackson, Sanders" right rbrace } {}
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Find the union of the sets F size 12{F} {} and B size 12{B} {} given as follows.

F = Aikman, Jackson, Rice, Sanders, Young size 12{F= left lbrace "Aikman, Jackson, Rice, Sanders, Young" right rbrace } {} B = Griffey, Jackson, Sanders, Thomas size 12{B= left lbrace "Griffey, Jackson, Sanders, Thomas" right rbrace } {}

The union of two sets is the set whose elements are either in A size 12{A} {} or in B size 12{B} {} or in both A size 12{A} {} and B size 12{B} {} . Therefore

F B = Aikman, Griffey, Jackson, Rice, Sanders, Thomas, Young size 12{F union B= left lbrace "Aikman, Griffey, Jackson, Rice, Sanders, Thomas, Young" right rbrace } {}

Observe that when writing the union of two sets, the repetitions are avoided.

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Let the universal set U = red, orange, yellow, green, blue, indigo, violet size 12{U= left lbrace "red, orange, yellow, green, blue, indigo, violet" right rbrace } {} , and P = red, yellow, blue size 12{P= left lbrace "red, yellow, blue" right rbrace } {} . Find the complement of P size 12{P} {} .

The complement of a set P size 12{P} {} is the set consisting of elements in the universal set U size 12{U} {} that are not in P size 12{P} {} . Therefore,

P ˉ = orange, green, indigo, violet size 12{ { bar {P}}= left lbrace "orange, green, indigo, violet" right rbrace } {}

To achieve a better understanding, let us suppose that the universal set U size 12{U} {} represents the colors of the spectrum, and P size 12{P} {} the primary colors, then P ˉ size 12{ { bar {P}}} {} represents those colors of the spectrum that are not primary colors.

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Questions & Answers

how do you translate this in Algebraic Expressions
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Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
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types of nano material
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I start with an easy one. carbon nanotubes woven into a long filament like a string
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Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
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after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
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name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
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this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
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Source:  OpenStax, Applied finite mathematics. OpenStax CNX. Jul 16, 2011 Download for free at http://cnx.org/content/col10613/1.5
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