13 October 2015Birkbeck College, U. London1 Introduction to Computer Systems Lecturer: Steve Maybank Department of Computer Science and Information Systems Autumn 2015 Week 3a: Boolean Operations

Recap: Binary Numbers In the standard notation for binary numbers a bit string such as 1001 stands for a sum of powers of 2: 1x2 3 +0x2 2 +0x2 1 +1x2 0 Binary[1001] and Decimal[9] are different names for the same number 13 October 2015Birkbeck College, U. London2

Recap: Conversion of Decimal to Binary Let m be a strictly positive integer Binary[b] = Decimal[m] m=2q+r b=c||r Binary[c] = Decimal[q] 26 October 2010Birkbeck College, U. London3

Recap: Binary Addition 13 October 2015Brookshear, Section 1.54 column: ===== There is a carry from column 1 to column 2 and from column 2 to column 3 In tests or examinations, always show the carries

Recap: Two’s Complement Two’s complement representations can be added as if they were standard binary numbers. 13 October 2015Brookshear, Section == ====

George Boole 13 October 2015Birkbeck College, U. London (Lincoln)-1864 (Ballintemple) 1849: publishes “An investigation into the laws of thought, on which are founded the mathematical theories of logic and probabilities”

Comparison of Numbers 13 October 2015Birkbeck College, U. London7 Symbol Description TrueFalse >greater than 5 > 15 > 10 >=greater than or equal to6 >= 6 1 >= 7 <less than 5 < 6 5 < 5 <=less than or equal to 2 <= 2 3 <= 2 ==equal to4 == 4 4 == 5 True and False are referred to as truth values

Examples of Boolean Statements 5>4 NOT(4==2) (5>4) AND (6>2) (1>4) OR (2>6) Note the use of NOT, AND and OR to create complicated statements out of simpler ones. 13 October 2015Brookshear, Section 1.18

Application Boolean statements are used to control the order in which the instructions (or statements) of a program are carried out. A Boolean statement is either True (1) or False (0). Example: x=4; if(x>6) then S(1) else S(2) 13 October 2015Birkbeck College, U. London9

Binary Boolean Operation A unary Boolean operation takes a truth value as input and outputs a truth value. Example: NOT A binary Boolean operation takes two truth values as input, and outputs a truth value. Examples: AND, OR, XOR 13 October 2015Brookshear, Section 1.110

13 October 2015Brookshire, Section Boolean Operations: AND, OR ABC C = A AND B ABC C = A OR B These are truth tables. 1 = True, 0 = False.

13 October 2015Brookshire, Section Boolean Operations: XOR, NOT AC C = NOT A 1 = True, 0 = False ABC C=A XOR B

13 October 2015Birkbeck College, U. London13 Boolean Statements Today is Thursday OR Today is Friday NOT[It is morning] He is in London AND He is in Leeds Each statement has a truth value. If the truth values of statements of A, B are known, then the truth values of (A OR B), NOT(A) and (A AND B) can be calculated.

13 October 2015Birkbeck College, U. London14 Combinations of Boolean Operations NOT(A AND B) A OR NOT(B) (A OR B)OR C A OR (B OR C) NOT(NOT(A) AND NOT(B))

13 October 2015Birkbeck College, U. London15 Application of a Boolean Operation 1. k=0 2. n=27 3. If 2 k+1 > n, Print(k); Halt 4. k=k+1 5. go to 3 Variables in this program: k, n

13 October 2015Birkbeck College, U. London16 While Loop Version 1. k = 0 2. n = While (2 k+1 <= n) 4. k = k+1 5. EndWhile 6. Print k 7. Halt

Boolean Operations for Addition 13 October 2015Birkbeck College, U. London17 x 2 x 1 y 1 ====== z 3 z 2 z 1

Binary Addition 13 October 2015Birkbeck College, U. London18 x 2 x 1 y 1 ====== z 3 z 2 z 1

13 October 2015Birkbeck College, U. London19 How Many Binary Boolean Operations? Answer 16. Why? A truth table with columns A, B, C has one row for each pair of values of A, B. This gives four rows. Each Boolean operation is defined by the entries in column C. The number of different choices for column C is 2x2x2x2 = 16.