Class 10: Set Cardinality, Induction

cs2102: Discrete Mathematics
University of Virginia, Fall 2017

See course site for notes:
https://uvacs2102.github.io

Transcript

1. Class  10: Set  Cardinality Introducing  Induction cs2102:  Discrete  Mathematics  |  F17 uvacs2102.github.io   David  Evans  |  University  of  Virginia
2. Plan Set  Cardinality Introducing  Induction PS4  Questions (Induction  Practice)
3. Alternate  Definition The  cardinality of  the  set ℕ" =       ∈ ℕ ∧ <  } is  .    
4. Alternate  Definition The  cardinality of  the  set ℕ" =       ∈ ℕ ∧ <  } is  .    If  there  is  a  bijection between  two   sets,  they  have  the  same  cardinality.
5. Example What  is  the   cardinality  of   {  1, 2, 3, 4, 6  } ?   The  cardinality of  the  set  " =       ∈ ℕ ∧ <  }  is  .    If  there  is   a  bijection between  two  sets,  they  have  the  same  cardinality.
6. Relations  and  Sizes function: ≤ 1  out injective: ≤ 1 in   total: ≥ 1  out surjective: ≥ 1 in If  there  is  a  surjective  relation  between   and    what  do  we  know  about  their  sizes?  
7. Relations  and  Sizes function: ≤ 1  out injective: ≤ 1 in   total: ≥ 1  out surjective: ≥ 1 in If  there  is  a  surjective  function  between   and    what  do  we  know  about  their  sizes?  
8. Relations  and  Sizes function: ≤ 1  out injective: ≤ 1 in   total: ≥ 1  out surjective: ≥ 1 in If  there  is  a  total  surjective  injective  function  between   and    what  do   we  know  about  their  sizes?  
9. Union  Cardinality What  is  the   cardinality  of   ∪ ?   The  cardinality of  the  set  " =       ∈ ℕ ∧ <  }  is  .    If  there  is   a  bijection between  two  sets,  they  have  the  same  cardinality.
10. Power  Set The  power  set  of  A  is  the  set  of  all  subsets  of  A. ∈ ⟺ ⊆
11. Power  Set The  power  set  of  A  is  the  set  of  all  subsets  of  A. ∈ ⟺ ⊆ Is  ∅ ∈ ?
12. Power  Set The  power  set  of  A  is  the  set  of  all  subsets  of  A. ∈ ⟺ ⊆ Is   ∈ ?
13. Induction  Principle
16. PS4  Questions (I  will  talk  about  problem  10.)
22. Charge • PS4  Due  Friday  (6:29pm) • Next  week:   – “Strong”  Induction  (MCS  5) – Induction  Practice