Topic: Logical operation.

Algebra of logic

Academic Subject: Medical Information Science

Faculty: Foreign Student Training (Medcine)

Module No 2: Medical knowledge and decision making in medicine and

dentistry.

Objectives• Understand the main logic operations

• Learn how to construct the table of truth

• Learn main logic rules

• Understand the relationship between

Boolean logic and digital computer

circuits

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Introduction

• In the latter part of the

nineteenth century, George

Boole incensed philosophers

and mathematicians alike

when he suggested that

logical thought could be

represented through

mathematical equations.

– How dare anyone suggest

that human thought could be

encapsulated and

manipulated like an

algebraic formula?Olenets Svitlana

Introduction

• In the middle of the twentieth

century, computers were commonly

known as “thinking machines” and

“electronic brains.”

– Many people were fearful of them.

• Nowadays, we rarely ponder the

relationship between electronic

digital computers and human logic.

Computers are accepted as part of

our lives.

– Many people, however, are still

fearful of them.

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• Computers, as we know

them today, are

implementations of

Boole’s Laws of Thought.

– John Atanasoff and

Claude Shannon were

among the first to see

this connection.

Introduction

The true-false nature of Boolean logic

makes it compatible with binary logic

used in digital computers.

Electronic circuits can produce Boolean

logic operations.

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Boolean Algebra

Boolean algebra is a mathematical system for the

manipulation of variables that can have one of two values.

– In formal logic, these values are:

“true” and “false”

– In digital systems, these values are:

“on” and “off”

1 and 0

“high” and “low”

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Introduction: PL?

• In Propositional Logic (a.k.a Propositional

Calculus or Sentential Logic), the objects are

called propositions

• Definition: A proposition is a statement that is

either true or false, but not both

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Types of statements

• A simple expression

• Expression of the structure can be simple or

composite.

• The content expression contained either a

communication or statement of the existing

world. This expression is simple. For

example, "the diagnosis of myocardial

infarction", "the patient is observed cardiac

arrhythmia"

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• Composite expression (logical function)

• From simple statements by ligaments AND,

OR and NOT are formed composite

expression, called logic functions. Simple

expression of which is formed a compound,

called logical arguments. The sentence "The

patient feels severe pain in the jaw, the

mouth does not close on their own, it's hard

to swallow and speak" is a composite

expression (logic function "AND").

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Alphabet logic statements• In modern logic is a special part of the complex

expression - the logic statements. In the logic

expression using an artificial language that has

such significant means (alphabet logic statements):

• Variable expression of logic - A, B, C, D, ...

(propositional variables) denote the simple

expression.

• Signs of logical connectors: Δ - conjunction ; ∇ -

disjunction ; ⇒ - implication ; ⇔ - equality; ⌐ -

negation;

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Propositions: Examples

• The following are propositions

– Today is Monday M

– The grass is wet W

– It is raining R

• The following are not propositions

– C++ is the best language Opinion

– When is the pretest? Interrogative

– Do your homework Imperative

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Introduction: Proposition

• Definition: The value of a proposition is

called its truth value; denoted by

– T or 1 if it is true or

– F or 0 if it is false

• Opinions, interrogative, and imperative are not

propositions

• Truth tablep

0

1

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Logical connectives• Connectives are used to create a compound

proposition from two or more propositions– Negation (denote or !)

– And or logical conjunction (denoted )

– Or or logical disjunction (denoted )

– XOR or exclusive or (denoted )

– Implication (denoted or )

Biconditional (denoted or )

We define the meaning (semantics) of the logical connectives using truth tables

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Logical Connective: Negation

• p, the negation of a proposition p, is also a

proposition

• Examples:

– Today is not Monday

– It is not the case that today is Monday, etc.

• Truth tablep p

0 1

1 0

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Logical Connective: Logical And

• The logical connective And is true only when both of the

propositions are true. It is also called a conjunction

• Examples

– It is raining and it is warm

– (2+3=5) and (1<2)

– Schroedinger’s cat is dead and Schroedinger’s is not dead.

• Truth tablep q pq

0 0

0 1

1 0

1 1

p q pq

0 0 0

0 1 0

1 0 0

1 1 1

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Logical Connective: Logical Or

• The logical disjunction, or logical Or, is true if one or

both of the propositions are true.

• Examples

– It is raining or it is the second lecture

– (2+2=5) (1<2)

– You may have cake or ice cream

• Truth table p q pq pq

0 0 0

0 1 0

1 0 0

1 1 1

p q pq pq

0 0 0 0

0 1 0 1

1 0 0 1

1 1 1 1

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Logical Connective: Exclusive Or

• The exclusive Or, or XOR, of two propositions is true when

exactly one of the propositions is true and the other one is false

• Example

– The circuit is either ON or OFF but not both

– Let ab<0, then either a<0 or b<0 but not both

– You may have cake or ice cream, but not both

• Truth table

p q pq pq pq

0 0 0 0

0 1 0 1

1 0 0 1

1 1 1 1

p q pq pq pq

0 0 0 0 0

0 1 0 1 1

1 0 0 1 1

1 1 1 1 0

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Logical Connective: Implication (1)

• Definition: Let p and q be two propositions. The

implication pq is the proposition that is false when

p is true and q is false and true otherwise

– p is called the hypothesis, antecedent, premise

– q is called the conclusion, consequence

• Truth tablep q pq pq pq pq

0 0 0 0 0

0 1 0 1 1

1 0 0 1 1

1 1 1 1 0

p q pq pq pq pq

0 0 0 0 0 1

0 1 0 1 1 1

1 0 0 1 1 0

1 1 1 1 0 1

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Logical Connective: Implication (2)

• The implication of pq can be also read as– If p then q

– p implies q

– If p, q

– p only if q

– q if p

– q when p

– q whenever p

– q follows from p

– p is a sufficient condition for q (p is sufficient for q)

– q is a necessary condition for p (q is necessary for p)

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Exercise: Which of the following implications is

true?

• If -1 is a positive number, then 2+2=5

• If -1 is a positive number, then 2+2=4

• If x = 0, then x+1 = 0

True. The premise is obviously false, thus no matter what the

conclusion is, the implication holds.

True. Same as above.

False

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Logical Connective: Biconditional (1)

• Definition: The biconditional pq is the

proposition that is true when p and q have the

same truth values. It is false otherwise.

• Note that it is equivalent to (pq)(qp)

• Truth tablep q pq pq pq pq pq

0 0 0 0 0 1

0 1 0 1 1 1

1 0 0 1 1 0

1 1 1 1 0 1

p q pq pq pq pq pq

0 0 0 0 0 1 1

0 1 0 1 1 1 0

1 0 0 1 1 0 0

1 1 1 1 0 1 1

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Logical Connective: Biconditional (2)

or equivalence

• The biconditional pq can be equivalently read as

– p if and only if q

– p is a necessary and sufficient condition for q

– if p then q, and conversely

– p iff q (Note typo in textbook, page 9, line 3)

• Examples

– x>0 if and only if x2 is positive

– The alarm goes off iff a burglar breaks in

– You may have pudding iff you eat your meat

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Truth Tables

• Truth tables are used to show/define the

relationships between the truth values of

– the individual propositions and

– the compound propositions based on them

p q pq pq pq pq pq

0 0 0 0 0 1 1

0 1 0 1 1 1 0

1 0 0 1 1 0 0

1 1 1 1 0 1 1

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Constructing Truth Tables

• Construct the truth table for the following

compound proposition

(( p q ) q )

p q pq q (( p q ) q )

0 0 0 1 1

0 1 0 0 0

1 0 0 1 1

1 1 1 0 1

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Precedence of Logical Operators

• As in arithmetic, an ordering is imposed on the use of logical operators in compound propositions

• However, it is preferable to use parentheses to disambiguate operators and facilitate readability

p q r (p) (q (r))

• To avoid unnecessary parenthesis, the following precedenceshold:

1. Negation ()

2. Conjunction ()

3. Disjunction ()

4. Implication ()

5. Biconditional ()

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Usefulness of Logic

• Logic is more precise than natural language

– You may have cake or ice cream.

• Can I have both?

– If you buy your air ticket in advance, it is cheaper.

• Are there or not cheap last-minute tickets?

• For this reason, logic is used for hardware and software specification

– Given a set of logic statements,

– One can decide whether or not they are satisfiable (i.e., consistent), although this is a costly process…

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Terminology:

Tautology, Contradictions, Contingencies

• Definitions– A compound proposition that is always true, no matter

what the truth values of the propositions that occur in it is called a tautology

– A compound proposition that is always false is called a contradiction

– A proposition that is neither a tautology nor a contradiction is a contingency

• Examples– A simple tautology is p p

– A simple contradiction is p p

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Logical Equivalences: Example 1

• Are the propositions (p q) and (p q)

logically equivalent?

• To find out, we construct the truth tables for

each: p q pq p pq

0 0

0 1

1 0

1 1

The two columns in the truth table are identical, thus we conclude that

(p q) (p q)

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Boolean Algebra

• Digital computers contain circuits that implement

Boolean functions.

• The simpler that we can make a Boolean function, the

smaller the circuit that will result.

– Simpler circuits are cheaper to build, consume less

power, and run faster than complex circuits.

• With this in mind, we always want to reduce our

Boolean functions to their simplest form.

• There are a number of Boolean identities that help us to

do this.

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Boolean Algebra

• Most Boolean identities have an AND (product) form as

well as an OR (sum) form. We give our identities using

both forms. Our first group is rather intuitive:

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Boolean Algebra

• Our second group of Boolean identities should be

familiar to you from your study of algebra:

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Boolean Algebra

• Our last group of Boolean identities are perhaps the most

useful.

• If you have studied set theory or formal logic, these laws

are also familiar to you.

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Boolean Algebra

• We can use Boolean identities to simplify the function:

as follows:

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• We have looked at Boolean functions in abstract terms.

• In this section, we see that Boolean functions are

implemented in digital computer circuits called gates.

• A gate is an electronic device that produces a result based

on two or more input values.

– In reality, gates consist of one to six transistors, but digital

designers think of them as a single unit.

– Integrated circuits contain collections of gates suited to a

particular purpose.

Logic Gates

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• The three simplest gates are the AND, OR, and NOT gates.

• They correspond directly to their respective Boolean operations, as you can see by their truth tables.

Logic Gates

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Logic Gates

• Gates can have multiple inputs and more than one output.

– A second output can be provided for the complement of the operation.

– We’ll see more of this later.

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Sequential Circuits

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Methods of presentation logic

functions• Logical functions (complex expression) can set in three ways: verbal, tabular

and analytical.

• I. In the verbal mode of presentation affects the function of words and

description should clearly identify all cases where logical arguments take

their possible values: F and T. For example, the function is and if any two

arguments equal and, in other cases F.

• II. Tabular method feeder logic function is a truth table. here with method,

using verbal descriptions, up table, which takes into account all possible

combination of values of logical arguments and function values for each

combination .

• III. Analytical method - a record of the logical functions as an equation that

reach out truth table. Output logic equation is of particular interest because

electronic circuits that apply in computing, built on the basis of pre-

compiled logic equations.Olenets Svitlana

Venn diagrams

• A Venn diagram is a representation of a Boolean

operation using shaded overlapping regions. There

is one region for each variable, all circular in the

examples here. The interior and exterior of

region x corresponds respectively to the values 1

(true) and 0 (false) for variable x. The shading

indicates the value of the operation for each

combination of regions, with dark denoting 1 and

light 0 (some authors use the opposite

convention).

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The AND operator

(both, all)

• rivers AND salinity

• dairy products AND export

AND Europe

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The OR operator

(either, any)

• fruit OR vegetables

• fruit OR vegetables OR cereal

from CSIRO AustraliaOlenets Svitlana

The NOT operator

• fruit NOT apples

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Write out logic statements using

Boolean operators for these.

• You have a buzzer in your car that sounds when

your keys are in the ignition and the door is open.

• You have a fire alarm installed in your house.

This alarm will sound if it senses heat or smoke.

• There is an election coming up. People are

allowed to vote if they are a citizen and they are

18.

• To complete an assignment the students must do a

presentation or write an essay.

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• Computers are implementations of Boolean logic.

• Boolean functions are completely described by truth

tables.

• Logic gates are small circuits that implement Boolean

operators.

• The basic gates are AND, OR, and NOT.

– The XOR gate is very useful in parity checkers and

adders.

Conclusion

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