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Computer Science 101 Boolean Algebra

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What’s next? A new type of algebra – Helps us A new type of algebra – Helps us With logical reasoningWith logical reasoning Understand and design circuits of a computerUnderstand and design circuits of a computer The “innards” of a computer The “innards” of a computer Basic circuitsBasic circuits Major components and how they work togetherMajor components and how they work together Low level instructions – machine languageLow level instructions – machine language How data and instructions are stored in computerHow data and instructions are stored in computer

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George Boole English mathematician English mathematician 1815-1864 1815-1864 1854: Introduction to the Laws of Thought 1854: Introduction to the Laws of Thought Boolean algebra Boolean algebra Logic Logic Set Theory Set Theory Circuits Circuits Programming: Conditions in “while” and “if” Programming: Conditions in “while” and “if”

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Boolean Constants and Variables In Boolean algebra, there are only two constants. In Boolean algebra, there are only two constants. True and False True and False On and Off On and Off +5v and 0v +5v and 0v 1 and 0 1 and 0 Boolean variables are variables that store values that are Boolean constants. Boolean variables are variables that store values that are Boolean constants.

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Boolean Operator AND If A and B are Boolean variables (or expressions) then A AND B is True (1) if and only if both A and B have values of True (1). If A and B are Boolean variables (or expressions) then A AND B is True (1) if and only if both A and B have values of True (1). We denote the AND operation like multiplication in ordinary algebra: AB or A. B We denote the AND operation like multiplication in ordinary algebra: AB or A. B

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Boolean Operator OR If A and B are Boolean variables (or expressions) then A OR B is True (1) if and only if at least one of A and B has value of True (1). If A and B are Boolean variables (or expressions) then A OR B is True (1) if and only if at least one of A and B has value of True (1). We denote the OR operation like addition in ordinary algebra: A+B We denote the OR operation like addition in ordinary algebra: A+B

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Boolean Operator NOT If A is a Boolean variable (or expression) then NOT A has the opposite value from A. If A is a Boolean variable (or expression) then NOT A has the opposite value from A. We denote the NOT operation by putting a bar over the variable (or expression) _ A We denote the NOT operation by putting a bar over the variable (or expression) _ A

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Boolean Expressions As with ordinary algebra, a Boolean expression is a well-formed expression made from As with ordinary algebra, a Boolean expression is a well-formed expression made from Boolean constants Boolean constants Boolean variables Boolean variables Operators AND, OR and NOT Operators AND, OR and NOT Parentheses Parentheses Example: _ ____ AB + (A+C)B Example: _ ____ AB + (A+C)B

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The value of a Boolean expression At any point, the value of a BE can be computed using the current values of the variables. At any point, the value of a BE can be computed using the current values of the variables. Unlike ordinary algebra, for a BE, there are only finitely many possible assignments of values to the variables; so, theoretically, we can make a table, called a truth table that shows the value of the BE for every possible set of values of the variables. Unlike ordinary algebra, for a BE, there are only finitely many possible assignments of values to the variables; so, theoretically, we can make a table, called a truth table that shows the value of the BE for every possible set of values of the variables.

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Truth Table: _ ____ E = AB + (A+C)B

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In Python!

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Laws of Algebra? In ordinary algebra, we have a distributive law : A(B+C) = AB + AC In ordinary algebra, we have a distributive law : A(B+C) = AB + AC What does it mean to say this is a law? What does it mean to say this is a law? The left side has parentheses, right side doesn’t.The left side has parentheses, right side doesn’t. The left side has one multiplication and the right side has two.The left side has one multiplication and the right side has two.

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Laws of Algebra? A(B+C) = AB + AC A(B+C) = AB + AC No matter what the numerical values of A, B, and C are, the two indicated computations will have the same value. No matter what the numerical values of A, B, and C are, the two indicated computations will have the same value.

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Laws of Boolean Algebra

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Boolean Expression Simplification

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