2 6 Infinite Sets Which set is larger

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 1 st number in the set, when n = 2 we get the 2 nd 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?

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.

• 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 (aleph null).

• To show a set has cardinal number : – 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,