Lecture 4

1. Lecture 4 Truth Tables of compound propositions

2. Lecture 3 : truth tables of Compound propositions. Objectives Understand the logic behind the definition of the compound statements . Construct truth tables for compound statements. Determine the true value of a compound statement for a specific case.

3. OVERVIEW By using logical connectives we can build up complicated compound proposition , and use the truth tables to determine the truth values of these compound proposition .

4. Example 1: Construct the truth table of the compound proposition (p V �q) ? (p ^ q)

5. Examples 2: For the below truth table we can observe that p ? q is a short form for (p? q) ? (q ? p)

6. Precedence of logical operators The table below represent the precedence Rule of logical operators i.e : p^q V r means (p^q)Vr pV q r means (pV q ) r

7. Translating English Sentences English sentences can be translated into into expression with propositional variables and logical connectives to :- Remove sentences ambiguity. Determine sentences truth values by analyze it�s logical expressions Use the rules of inference to reasoning these sentences .

8. Example 1 Let p and q ,be the propositions p = �It is below freezing� q = �It is snowing � Write these propositions using p and q and logical connectives . 1- It is below freezing and it is snowing (p ^ q) 2-It is below freezing but not snowing (p V�q) 3- It is not below freezing and it is not snowing (� p^�q ) 4-It is either snowing or below freezing (or both) (p V q ) 5-If it is below freezing, it is also snowing (p q) 6- It is either below freezing or it is snowing, but it is not snowing if it is below freezing ((p Vq) ^ (p^�q)) 7- That it is below freezing is necessary and sufficient for it to be snowing or (i.e it snowing iff it is below freezing) q p

9. Example 2 You can access the Internet from campus only if you are a computer science major or you are not a freshman. write with proposition a,c and f and logical connective: Let a : access the internet c: computer science student f: not a fresh man a (c V � f) You cannot ride the roller coaster if you are under 4 feet tall unless you are older than 16 years old. p: you can ride the roller coaster q: you are under 4 feet tall s: you are older that 16 years old ( q ^ �s) �p

10. Logical and bit operations A bit is a symbol with two only two values 0 and 1 referred to as binary digit . A bit can used to represent a truth values :1 bit to true , 0 bit to false . A Boolean variables can be represented using a bits as it takes only either true or false .

11. Bit operations Bit operations are correspond to logical connectives by using 1 for T , 0 for F in truth tables . OR,AND,XOR are used instead of ^,V, Respectively. Information represented using bit string which is a sequence of zeros and ones , length of this string is the number of bits in the string .

12. Bit operators OR,AND,XOR

13. Bitwise OR,AND and XOR To find the bitwise OR ,AND and XOR of strings, we apply the truth values for each bit with it�s corresponding bit in other bit string . Ex: find the bitwise and for strings :11101010 and 11001101 Solution:- Arrange element as blocks of size 4 bit (right to left) 1 1 1 0 1 0 1 0 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 bitwise OR 1 1 0 0 1 0 0 0 bitwise AND 0 0 1 0 0 1 1 1 bitwise XOR

14. More Examples find the bitwise OR ,AND and XOR of string of a ) 1011110 and 0100001 1 0 1 1 1 1 0 0 1 0 0 0 0 1 1 1 1 1 1 1 1 Bitwise Or 0 0 0 0 0 0 0 Bitwise And 1 1 1 1 1 1 1 Bitwise XOR

15. More Examples b) b) 0 1 0 1 0 1 1 0 and 0 0 1 1 0 0 1 0 0 1 0 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 0 1 0 0 1 0 Bitwise AND 0 1 1 1 0 1 1 0 Bitwise OR 0 1 1 0 0 1 0 0 Bitwise XOR