Lecture 4

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Lecture 3 : truth tables of Compound propositions.. ObjectivesUnderstand 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.. OVERVIEW. By using logical connectives we

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Lecture 4

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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

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