Download Set Concepts

Section 2.1 Set Concepts
Set
- a collection of objects. Other words for objects are elements or members.
Ex.: the set of majors at ECC
Ex.: the set of pets living in my house.
Well defined set
- a set that is described in such a way that it is completely clear about what is in it.
Ex.: the set of songs written by Chantal Kreviazuk is well defined.
Ex.: the set of the most generous Americans is not well defined. The people whom you think are
generous, I may disagree. Opinion plays a role.
the Natural Numbers
N = {1, 2, 3, ...} (Also called the counting numbers.)
The … is called an ellipsis, a symbol meaning "and so on", or "follow in similar fashion"
Another example: {3, 4, 5, ... 22} is the Natural numbers between 2 and 23. (Or you can say “the
Natural numbers between 3 and 22, inclusive".)

- is read “is an element of ”
Ex.: 7  {1, 2, 3, …} or equivalently, 7  N.
Ex.: @  {2, *, $, @ }
3 ways to represent sets:
1. Description in words
2. Roster – list the elements, separate with commas, place { }’s around them
a. Note, for example, that {1, 2, 3} = {3, 2, 1, } – order does not matter, and
b. Note, for example, that {a, a, b} = {a, b}-- do not repeat same elements.
c. A set may be labeled with a capital letter. Ex: D = {1, 2, 3}
3. Set builder notation – {x | x …describe condition(s) on x }
 the first x is read “all of the elements x” or simply “x”
 the | is read “such that”
Ex.: {x | x  N and x < 5} is read “All of the elements x such that x is an element of the
Naturals (N ) and x is less than 5.”
Ex.: {x | x is the name of a state of the United States} read “All of the elements x such that
x is the name of a state of the United States”
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
{x | x
{x | x
{x | x
{x | x
Review of inequality symbols:
 N and x < 5 } in roster form is {1, 2, 3, 4}
 N and x > 5 } in roster form is {6, 7, 8, …}
 N and x ≤ 5 } in roster form is {1, 2, 3, 4, 5}
 N and x ≥ 5 } in roster form is {5, 6, 7, 8, …}
Cardinal Number of set A
- symbolized as n(A) is the number of elements in the set, or the size of the set
Ex.: If A = {do, re, mi}, then n(A) = 3, “the cardinal number of A is 3”.
Ex.: If C = {3, …, 10}, then n(C) = 8.
Finite set
- a set whose cardinal number is a Natural number.
Ex.: If A = {^, *, $, #, 4, T, R}, then n(A) = 7. Since 7  N, A is a finite set.
- (A set that “stops”, that you can count, speaking informally.)
Infinite set
- a set that is not finite
Ex.: the set of Rational numbers (numbers that can be expressed as fractions)
Equivalent sets:
-- set A is equivalent to set B iff (if and only if or “means”) n(A) = n(B). In other words iff the
number of elements in A is equal to the number of elements in B.
Ex.: If A = {1, 2, 3} and B = {K, L, a}
Equal (=) sets:
-- if set A and B are exactly the same set, then A = B.
Ex.: If A = {1, 2, 3} and B = {3, 1, 2} then they ARE the same set and A = B.
One-to-one correspondence
- Two sets with the same number of elements (cardinal number) can be lined up in a correspondence
such that each element of the first set is matched with an element in second set.
Empty set (null set)
- symbolized { } or Ǿ. The set with no elements
Ex.: {x | x  N and x + 2 = 0}
Universal set
- symbolized U. The set of all elements under consideration
Ex.: In a math text book the directions might read: “Solve the equations where you answer may be
any Real number. Then the Real numbers, in this context, is the universal set.
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