B3 seminar

1. B3 Seminar
(1/13/2015)
Tamura Takumi

2. Abstract of text (pages 1-8)
Discrete mathematics has quite a wide range

3. Floor and Ceiling function (page 9)
 Floor function 𝑥 ≔ 𝑚𝑎𝑥 𝑛 ∈ ℤ | 𝑛 ≤ 𝑥
 Example: 3.68 = 3, 0.999 = 0
 Ceiling function 𝑥 ≔ 𝑚𝑖𝑛 𝑛 ∈ ℤ | 𝑛 ≥ 𝑥
 Example: 3.68 = 4, 0.999 = 1

4. Set operations (pages 10-15)
 Cartesian product 𝐴 × 𝐵
 𝐴 × 𝐵 = 𝑎, 𝑏 | 𝑎 ∈ 𝐴 𝑎𝑛𝑑 𝑏 ∈ 𝐵
Be careful !
𝐴 × 𝐵 ≠ 𝐵 × 𝐴
∵ 𝐴 × 𝐵 is set of ordered pair.

5. Set operations (pages 10-15)
 Union sets A ∪ 𝐵
 𝐴 ∪ 𝐵 = 𝑥 | 𝑥 ∈ 𝐴 𝑜𝑟 𝑥 ∈ 𝐵
Point !
𝐴 ⊂ 𝐴 ∪ 𝐵 ⊂ 𝐴 ∪ 𝐵 ∪ 𝐶 ⊂ ⋯

6. Set operations (pages 10-15)
 Intersection of sets A ∩ 𝐵
 𝐴 ∩ 𝐵 = 𝑥 | 𝑥 ∈ 𝐴 𝑎𝑛𝑑 𝑥 ∈ 𝐵
Point !
𝐴 ⊃ 𝐴 ∩ 𝐵 ⊃ 𝐴 ∩ 𝐵 ∩ 𝐶 ⊃ ⋯

7. Set operations (pages 10-15)
 Power set of 𝐴=2 𝐴
(= 𝒫 𝐴 )
 2 𝐴
= 𝑋 | 𝑋 ⊂ 𝐴
 𝐴 ∈ 2 𝐴
 if 𝐴 is finite set then 2 𝐴
= 2 𝐴

8. Exercise