1. Introduction to Computer Systems
Department of Computer Science and Information Systems
Lecturer: Steve Maybank
Spring 2020
Week 3a: Boolean Operations
28 January 2020
Birkbeck College, U. London

2. Recall: Binary Numbers
In the standard notation for binary numbers a bit string such as 1001 stands for a sum of powers of 2: 1x23+0x22+0x21+1x20
Binary[1001] and Decimal[9] are different names for the same number
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3. Recall: Conversion of Decimal to Binary
Let m be a strictly positive integer
Problem: find b such that Binary[b] = Decimal[m]
Solution:
m = 2q+r # r is the rightmost bit in b
Binary[c] = Decimal[q] # easier because q < m
b = c||r
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4. Recall: Binary Addition
column:
1 1
=====
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
Brookshear, Section 1.5

5. Recall: Two’s Complement
If 𝑥,𝑦, 𝑥+𝑦 are in the correct range, then the two’s complement representations of 𝑥 and 𝑦 can be added as if they were standard binary numbers to obtain the tc representation of 𝑥+𝑦.
TC 𝑥 = rightmost n bits of 𝑥+ 2 𝑛
TC 𝑥+𝑦 =rightmost_n 𝑥+𝑦+ 2 𝑛
=rightmost_n 𝑥+ 2 𝑛 +𝑦+ 2 𝑛
=rightmost_n(TC(𝑥)+TC(y))
Brookshear, Section 1.6

6. Worksheet for Week 2 Convert binary 1011 to decimal. Convert decimal
1011 to binary.
Convert hexadecimal 12F and 194 to binary
Attempt the decimal sums 5+1, 12+5, 5-11 using
5 bit two’s complement.
If 𝑠 is a bit string such that
reverse 𝑠 =complement 𝑠
then show that the length of 𝑠 is even
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7. Birkbeck College, U. London
George Boole
1815 (Lincoln)-1864 (Ballintemple)
1849: publishes “An investigation into
the laws of thought, on which are
founded the mathematical theories
of logic and probabilities”
Birkbeck College, U. London

8. Birkbeck College, U. London
Comparison of Numbers
 Symbol
 Description
 True
False 
 >
greater than 
 5 > 1
5 > 10 
 >=
greater than or equal to
6 >= 6 
 1 >= 7
 <
less than 
 5 < 6
 5 < 5
 <=
less than or equal to
 2 <= 2
 3 <= 2
 ==
equal to
4 == 4 
 4 == 5
True and False are referred to as truth values
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9. 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
Brookshear, Section 1.1

10. Birkbeck College, U. London
Application
A Boolean statement is either True (1) or False (0)
Boolean statements are used to control the order in which the instructions in a program are carried out
Example: x = 4
if(x > 6) then code_A else code_B
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11. Unary and Binary Boolean Operations
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
Brookshear, Section 1.1

12. Boolean Operations: AND, OR
These are truth tables. 1 = True, 0 = False. A B C 1 A B C 1
C = A AND B
C = A OR B
Brookshire, Section 1.1

13. Boolean Operations: XOR, NOT
1 = True, 0 = False A B C 1 A C 1
C = NOT A
C=A XOR B
Brookshire, Section 1.1

14. Birkbeck College, U. London
Boolean Statements
Today is Thursday OR Today is Friday
NOT[It is morning]
He is in London AND There is life on Mars
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.
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15. 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))
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16. Application of a Boolean Operation
k = 0
n = 27
If 2k+1 > n, Print(k); halt
k = k+1
go to 3
Variables in this program: k, n
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17. Birkbeck College, U. London
While Loop Version
k = 0
n = 27
while (2k+1 <= n)
k = k+1
endWhile
print k
halt
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18. Boolean Operations for Addition
Each digit z 1 , 𝑧 2 , 𝑧 3
can be obtained using
a ternary Boolean
operation,
𝑧 1 = 𝑜𝑝 1 𝑥 1 , 𝑥 2 , 𝑦 1
𝑧 2 = 𝑜𝑝 2 𝑥 1 , 𝑥 2 , 𝑦 1
𝑧 3 = 𝑜𝑝 3 𝑥 1 , 𝑥 2 , 𝑦 1
x2 x1 y1
======
z3 z2 z1
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19. Birkbeck College, U. London
Binary Addition
x2 x1 y1
======
z3 z2 z1
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20. 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.
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21. Birkbeck College, U. London
Example
Use truth tables to show that
A OR (B AND C) = (A OR B) AND (A OR C)
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