Download 2.6 Infinite Sets

2.6 Infinite Sets
• Which set is larger, the set of even counting
number or the set of odd counting numbers?
E = { 1, 2, 3, 4, 5, 6, 7, ……}
O = { 1, 3, 5, 7, 9, 11, 13…..}
• How can we determine which set is larger?
• Def: An infinite set is a set that can be placed in a
one-to-one correspondence with a proper subset of
itself.
• Showing a Set is Infinite
– Place the set in a one-to-one correspondence with a
proper subset of itself
– Show the pairing of the general terms in the sets
• General Terms: the general term in any set should be written in
terms of n such that when n = 1, we get the 1st number in the set, when
n = 2 we get the 2nd number in the set, etc.
– Counting numbers = { 1, 2, 3, 4,…., n ,….}
• Write the general term for this set of numbers.
• Find what the numbers differ by
• + or minus some number
{ 4, 9, 14, 19,….}
Pg. 93: In exercises 3-12 show that the set is infinite by placing it
in a one-to-one correspondence with a proper subset of itself. Be
sure to show the pairing of the general terms in the sets.
• HW: pg. 93 #’s 4, 5, 6, 8, 10, & 12
From section 2.1:
• How can you determine the cardinal
number of a finite set?
A  { # , ?, $ }
Countable Sets
• Def: A set is countable if it is finite or if it can be placed in a
one-to-one correspondence with the set of counting numbers.
A  { # , ?, $ }
B  {1, 2, 3}
• How do you determine the cardinal number
of an infinite set?
– All infinite sets that can be placed in a one-toone correspondence with the set of counting
numbers has cardinality 0 (aleph null).
• To show a set has cardinal number 0 :
– Establish a one-to-one correspondence between the set of
counting numbers and the given set
– Show the pairing of the general terms in the set
• HW :pgs. 93 #’s 13-16 all, 18-21 all,