Great Ideas Alan Turing – What is computable? A task is computable if one can specify a sequence of instructions which when followed will result in the completion of the task. John Von Neumann – Stored program concept Developed the concept of storing a program in the computer’s memory rather than it’s circuitry Copyright © 2008 by Helene G. Kershner

Great Ideas Technological advances Vacuum tube  transistor  integrated circuits (multiple transistors on a chip)  VLSI (very large scale integration) Microprocessor chip – computer on a chip High level programming language – people can build “user friendly” software” Changes in memory device hardware Storing information digitally, making use of the concept that computers work with two states: on/off, 1/0, yes/no, high current/low current The computer is a binary machine Copyright © 2008 by Helene G. Kershner

Great Ideas The computer is a binary machine Perform mathematics Perform operations on letters that form words Represent images in grayscale and color Represent sound Copyright © 2008 by Helene G. Kershner

The computer is a binary machine Copyright © 2008 by Helene G. Kershner

Boolean Logic / Boolean Algebra Boolean Algebra (Boolean Logic) is an algebra for symbolically representing problems in logic & analyzing them mathematically. Based on work of George Boole English mathematician An Investigation of the Laws of Thought Published in 1854 Reduced logic of human thought to mathematical operations An analysis of how natural language works if it were logical Copyright © 2008 by Helene G. Kershner

Boolean Logic / Boolean Algebra In 1938 Claude E. Shannon of MIT Showed that Boolean logic could be applied to the design of relay networks in telephone systems (the ability to switch signals from one place to another automatically) Invented branch of mathematics called Information Theory Information Theory applied Boolean algebra to certain engineering problems systems Today, boolean algebra, as applied to computer hardware design is also known as Switching Theory. Copyright © 2008 by Helene G. Kershner

Boolean Logic / Boolean Algebra Boolean Logic is an abstraction. Definition -- Abstraction: Is the process of taking away or removing characteristics from something in order to reduce it to a set of essential characteristics (Whatis.com) not concrete: not relating to concrete objects but expressing something that can only be appreciated intellectually (Encarta.msn.com) Considered apart from concrete existence: an abstract concept ( Copyright © 2008 by Helene G. Kershner

Boolean Logic / Boolean Algebra Applying Boolean Logic to computers allows them to handle very complex problems using complicated connections of simple components. Designing hardware and software is very complicated because we are using them to deal with complex tasks. Copyright © 2008 by Helene G. Kershner

Boolean Logic / Boolean Algebra Abstraction allows the designer to be separated from the machine to get a clearer picture of what needs to be done. allows us to drive a car without knowing how the engine works allows computer users to work with the hardware without knowing exactly how the machine does what it does. Add a flash stick, the software will recognize it Plug in a digital camera and the software will find it Move music onto my iPod from the hard-drive without having any real idea what the computer is doing.

Computers: Complex tasks with Simple Components Basic Computer Components Switches On/off, high current / low current, 1/0 Connectors Wires that connect switches – pipes Must be able to branch, one or more paths Copyright © 2008 by Helene G. Kershner

Boolean Logic / Boolean Algebra Logic Gates Hardware interpretation of boolean logic Universal building blocks, special circuits that perform the operation of: AND OR NOT From these basic elements, computer engineers can design “anything” Copyright © 2008 by Helene G. Kershner

Boolean Logic / Boolean Algebra Different and equal ways to represent this kind of logic Using the English works AND, OR, NOT Using Mathematical symbols Λ, ۰ (means AND) V, + (means OR) ~, ¯ (means NOT) Copyright © 2008 by Helene G. Kershner

Boolean Logic / Boolean Algebra Different and equal ways to represent this kind of logic Engineering symbols inverter/flipper Copyright © 2008 by Helene G. Kershner

Boolean Logic / Boolean Algebra Look at statements that can be determined to be true or false. I think therefore I am. This can be either true or false Yesterday my daughter in college called needing money. This can be either true or false My son lives in Washington DC. This can also be either true or false Copyright © 2008 by Helene G. Kershner

Boolean Logic / Boolean Algebra Look at statements that cannot be to be true or false. A Question is not a statement: What time is it? While an answer to a question may be Yes or No, the statement is neither true or false: Is it 3 o’clock? A command is not a statement: You MUST get an A in this class! Turn at the next corner. Wishes are not statements: I’d like to make a million dollars. Have a Happy Thanksgiving. Copyright © 2008 by Helene G. Kershner

Boolean Logic / Boolean Algebra Different and equal ways to represent this kind of logic Truth Tables Visually shows true/false values and the results (inputs and outputs) of a logic example. Describes what is happening in a logic gate or logic statement Uses T/F, 1/0 Copyright © 2008 by Helene G. Kershner

Boolean Logic / Boolean Algebra Logic can be applied to statements. Statements have the property of either being True (T, 1) or False (F, 0). Not everything we say is a statement because it cannot be thought of as having the value of either True or False. Copyright © 2008 by Helene G. Kershner

Venn Diagrams: A, B Boolean Logic The set A is mutually exclusive from set B They have nothing in common Copyright © 2009 by Helene G. Kershner

Venn Diagrams: A AND B Boolean Logic The set A and the set B have a common area. The pink area labeled 2, is represented by A AND B A B Copyright © 2009 by Helene G. Kershner

Logic -- AND AND combines two statements/inputs either one of which can be True or False A statement using AND is only true if both parts of the sentence are true John has a sub-prime mortgage AND John cannot pay his mortgage. Copyright © 2009 by Helene G. Kershner

Logic -- AND John has a sub-prime mortgage AND John cannot pay his mortgage. Truth table identifies the possibilities for AND: sub-prime mortgage AND cannot pay mortgage Result F F F F T F T F F T T T Copyright © 2009 by Helene G. Kershner

Logic -- AND A: sub-prime mortgage B: cannot pay mortgage A AND B Result F F F F T F T F F T T T Copyright © 2009 by Helene G. Kershner

Venn Diagram: A OR B Boolean Logic A OR B represents the area covered by all of A as well as all of B. A B Copyright © 2009 by Helene G. Kershner

Logic – OR OR combines two statements/inputs either one of which can be True or False A statement using OR is true if either parts of the sentence is true. I like mango OR chocolate ice cream. John has a sub-prime mortgage OR John has a traditional mortgage. John can afford his mortgage OR John is unable to pay his mortgage.. John has a sub-prime mortgage OR John is unable to pay his mortgage. Copyright © 2009 by Helene G. Kershner

Logic – OR I like mango OR chocolate ice cream. John has a sub-prime mortgage OR John has a traditional mortgage. John can afford his mortgage OR John is unable to pay his mortgage.. John has a sub-prime mortgage OR John is unable to pay his mortgage. A OR B A OR B R F F F F T T T F T T T T Copyright © 2008 by Helene G. Kershner

Logic – OR A: John has a sub-prime mortgage B: John is unable to pay his mortgage. A OR B Result A OR B R F F F F T T T F T T T T Copyright © 2009 by Helene G. Kershner

Venn Diagram: NOT A Boolean Logic NOT A is the same as ~ A NOT A is everything that is outside of A. A B Copyright © 2009 by Helene G. Kershner

Logic – NOT/Invert The NOT command, flips or Inverts the value it is given. The operation switches that statement between True and False I have a traditional mortgage. I do NOT have a sub-prime mortgage. NOT a sub-prime mortgage INVERTs or is the opposite sub-prime mortgage which is a traditional mortgage. A = I eat peanut-butter NOT A means I do NOT eat peanut-butter If A is True, then NOT A is False If A is False, then NOT A is True A = I love peanut-butter For me A is False, so NOT A would be True Copyright © 2009 by Helene G. Kershner

Logic – NOT/Invert Truth Table A NOT A = ~A Copyright © 2009 by Helene G. Kershner

Logic Symbols – Order of Operations Order of Operations: In a logic statement using the three operators we have learned the order is Parenthesis () NOT  NOT, ~ ¯¯ AND  AND, ۰, Λ OR  OR, +, V Copyright © 2009 by Helene G. Kershner

Venn Diagram: NOT (A AND B) Boolean Logic NOT (A OR B) is the same as area labeled 4. It is the area that is NOT or outside of (A OR B). AB Copyright © 2009 by Helene G. Kershner

VENN Diagrams: NOT (A OR B) Boolean Logic NOT (A AND B) is the pink area. (A AND B) is the white area, so NOT (A AND B) is all the area outside of (A AND B). A B Copyright © 2009 by Helene G. Kershner

Using Boolean Algebra in the “Real World” Boolean Operators

Logic Logic problems can relate back to English statements: A newly constructed building has three types of security alarms. It has an intrusion alarm, a fire alarm and a flood alarm. If the intrusion or fire alarm goes off then the police department is called. If the fire or flood alarm goes off then the fire department is called. But if the flood alarm goes off don’t call the police. And if the intrusion alarm goes off don’t call the fire department. A = intrusion alarm A OR B = Police B = fire alarm C = flood alarm B OR C = Fire Department Copyright © 2008 by Helene G. Kershner