1. CSE 20 DISCRETE MATH Prof. Shachar Lovett http://cseweb.ucsd.edu/classes/wi15/cse20-a/ Clicker frequency: CA

2. Todays topics Algorithms: number systems, binary representation Boolean logic Sections 1.3, 3.2-3.4 in Jenkyns, Stephenson

3. Operations on binary numbers Add, subtract, multiply,… … first, how do we add? A. 111 B. 100 C. 1011 D. 1111 E. None of the above

4. One bit addition 1 0 1 + 1 1 0 1

5. One bit addition 1 0 1 + 1 1 0 1

6. One bit addition 1 0 1 + 1 1 0 1 Carry: 0 1 1

7. One bit addition 1 0 1 + 1 1 0 1 Carry: 1 0 1 1

8. How to add binary numbers? 1 1 0 0 1 0 1 0 1 + 1 0 1 1 0 1 1 0 1 ? ? ? ? ?

9. How to add binary numbers? ? ? ? ? ? ? ? ? (carry) 1 1 0 0 1 0 1 0 1 + 1 0 1 1 0 1 1 0 1 ? ? ? ? ?

10. How to add binary numbers? Two basic operations: One-Bit-Addition(bit1, bit2, carry) Next-carry(bit1, bit2, carry) ? ? ? ? ? ? ? ? (carry) 1 1 0 0 1 0 1 0 1 + 1 0 1 1 0 1 1 0 1 ? ? ? ? ?

11. Numbers … logic … circuits

12. One bit addition One-Bit-Addition(bit1, bit2, carry) Bit1bit2carryOne-Bit-Addition FFFF FFTT FTFT FTTF TFFT TFTF TTFF TTTT T=1 F=0 Can we build this from “basic primitives”?

13. Logical operators JS pp. 82-84 Truth table is the definition of the operator! PQ P  QP  Q TTT TFF FTF FFF PQP v Q TTT TFT FTT FFF P~P TF FT

14. Logical operators JS pp. 82-84 PQ P  QP  Q TTT TFF FTF FFF PQP v Q TTT TFT FTT FFF P~P TF FT PQP XOR Q TT? TF? FT? FF? A.T, T, T, F B.T, F, F, T C.F, T, F, T D.F, F, F, T E.None of the above.

15. OR vs XOR A OR B: Either A, or B, or both A XOR B: Either A, or B, but not both In spoken language, we sometimes confuse them In mathematics, we needs to be precise

16. OR vs XOR A OR B: Either A, or B, or both A XOR B: Either A, or B, but not both In spoken language, we sometimes confuse them In mathematics, we needs to be precise You are at a restaurant. The menu says you can have either a salad or a soup. Mathematically, this is: A. OR B. XOR C. Both D. Neither

17. OR vs XOR A OR B: Either A, or B, or both A XOR B: Either A, or B, but not both In spoken language, we sometimes confuse them In mathematics, we needs to be precise You are at a birthday. The host asks: do you want ice-cream or cake (you can have both). Mathematically, this is: A. OR B. XOR C. Both D. Neither

18. Boolean expressions JS p. 83 How to translate to a truth table? Is there a unique way? A. Yes B. No (P  Q)  (~R)

19. Boolean expressions How to translate to a truth table? How many rows (not including header)? A. 2 B. 3 C. 4 D. 8 (P  Q)  (~R)

20. Boolean expressions PQR PQPQ ~R (P  Q)  (~R) FFF FFT FTF FTT TFF TFT TTF TTT

21. Boolean expressions PQR PQPQ ~R (P  Q)  (~R) FFFF FFTF FTFF FTTF TFFF TFTF TTFT TTTT

22. Boolean expressions PQR PQPQ ~R (P  Q)  (~R) FFFFT FFTFF FTFFT FTTFF TFFFT TFTFF TTFTT TTTTF

23. Boolean expressions PQR PQPQ ~R (P  Q)  (~R) FFFFTT FFTFFF FTFFTT FTTFFF TFFFTT TFTFFF TTFTTT TTTTFT

24. Truth table to expression PQR??? TTTT TTFT TFTT TFFF FTTF FTFF FFTT FFFF

25. DNFs DNF = Disjunctive Normal Form Literal: variable or its negation: P, ~P Term: AND of literals: P  ~Q DNF: OR of terms: (P  ~Q)  (R  P)  ~P Theorem: any Boolean expression can be written as a DNF

26. Truth table to expression PQR??? TTTT TTFT TFTT TFFF FTTF FTFF FFTT FFFF

27. Truth table to expression PQR??? TTTT TTFT TFTT TFFF FTTF FTFF FFTT FFFF PQRPQR P  Q  ~R P  ~Q  R ~P  ~Q  R (P  Q  R)  (P  Q  ~R)  (P  ~Q  R)  (~P  ~Q  R)

28. Truth tables and Circuits T“1”High voltage F“0”Low voltage

29. 4 bit adder

30. Next class Quantifiers and paradoxes Read sections 3.2-3.4 in Jenkyns, Stephenson Google “liar’s paradox”