Introduction to Discrete Mathematics: Counting Theory and Combinatorics, Slides of Discrete Mathematics

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Introduction to Discrete

Mathematics

If I have 14 teeth on the top and 12

teeth on the bottom, how many

teeth do I have in all?

Addition Rule (2 possibly overlapping sets)

Let A and B be two finite sets

|A∪B| =

|A| + |B| - |A∩B|

Addition of multiple disjoint sets:

• Let A 1 , A 2 , A 3 , …, An be disjoint, finite sets.

A i Ai

i=

n

i

n

=

1

S = all possible outcomes of one white die and one black die.

Partition Method

S = all possible outcomes of one white die and one black die.

Partition S into 6 sets:

Partition Method

• A 1 = the set of outcomes where the white die is 1.
• A 2 = the set of outcomes where the white die is 2.
• A 3 = the set of outcomes where the white die is 3. A 4 = the set of outcomes where the white die is 4. A5 = the set of outcomes where the white die is 5. A6 = the set of outcomes where the white die is 6.

Each of 6 disjoint sets have size 6 = 36 outcomesDocsity.com

• Ai ≡ set of outcomes where black die says i and the white die says something else.

S A (^) i A (^) i 5 30 i=1 i=

= = = =

∑ ∑ i 1

6 6 6 

S ≡ Set of all outcomes where the dice show different values. S =?

| S ∪ T | = # of outcomes = 36 |S| + |T| = 36 |T| = 6 |S| = 36 – 6 = 30

S ≡ Set of all outcomes where the dice show different values. S =?

T ≡ set of outcomes where dice agree. = { <1,1>, <2,2>, <3,3>,<4,4>,<5,5>,<6,6>}

S ≡ Set of all outcomes where the black die shows a smaller number than the white die. S =?

Ai ≡ set of outcomes where the black die says i and the white die says something larger.

S = A 1 ∪ A 2 ∪ A 3 ∪ A 4 ∪ A 5 ∪ A 6

|S| = 5 + 4 + 3 + 2 + 1 + 0 = 15

It is clear by symmetry that | S | = | L |.

S + L = 30

Therefore | S | = 15

S ≡ Set of all outcomes where the black die shows a smaller number than the white die. S =?

L ≡ set of all outcomes where the black die shows a larger number than the white die.

S L

Pinning Down the Idea of Symmetry by Exhibiting a Correspondence

Put each outcome in S in correspondence with an outcome in L by swapping color of the dice.

Thus: S = L

Each outcome in S gets matched with exactly one outcome in L, with none left over.

• f is 1-1 if and only if
• ∀x,y∈A, x ≠ y ⇒ f(x) ≠ f(y)

For Every

There Exists

f is onto if and only if

∀z∈B ∃x∈A f(x) = z

Let f : A → B Be a Function From a Set A to a Set B

A B

∃ onto f : A → B ⇒ | A | ≥ | B |

A B

∃ 1-1 onto f : A → B ⇒ | A | = | B |