Deriving the Formula for the Sum of a Geometric Series

... chunks", I promise to supply the formula for the sum of a geometric series and the mathematical derivation of it. Here it is. Consider a sum of terms each of which is a successively higher power of a number or an algebraic quantity represented by a variable: 1 + x + x 2 + ... + x n . Note that the f ...

... chunks", I promise to supply the formula for the sum of a geometric series and the mathematical derivation of it. Here it is. Consider a sum of terms each of which is a successively higher power of a number or an algebraic quantity represented by a variable: 1 + x + x 2 + ... + x n . Note that the f ...

Document

... The duality principle is formed by replacing AND with OR, OR with AND, 0 with 1, 1 with 0, variables and complements are left unchanged. de Morgan’s laws Allow us to convert between types of gates; we can generalize them to n variables: ...

... The duality principle is formed by replacing AND with OR, OR with AND, 0 with 1, 1 with 0, variables and complements are left unchanged. de Morgan’s laws Allow us to convert between types of gates; we can generalize them to n variables: ...

1. Binary operators and their representations

... Boolean algebra is the basic mathematics needed for logic design of digital systems; Boolean algebra uses Boolean (logical) variables with two values (0 or ...

... Boolean algebra is the basic mathematics needed for logic design of digital systems; Boolean algebra uses Boolean (logical) variables with two values (0 or ...

... The duality principle is formed by replacing AND with OR, OR with AND, 0 with 1, 1 with 0, variables and complements are left unchanged. de Morgan’s laws Allow us to convert between types of gates; we can generalize them to n variables: ...

A + B + C

... In POS standard form, every variable in the domain must appear in each sum term of the expression. You can expand a nonstandard POS expression to standard form by adding the product of the missing variable and its complement and applying rule 12, which states that (A + B)(A + C) = A + BC. Convert X ...

... In POS standard form, every variable in the domain must appear in each sum term of the expression. You can expand a nonstandard POS expression to standard form by adding the product of the missing variable and its complement and applying rule 12, which states that (A + B)(A + C) = A + BC. Convert X ...

CS302midtermpaper

... 6. the diagram above shows the general implementation of ________ form * POS 7. The expression _________ is an example of Commutative Law for Multiplication. * AB=BA (Page 72) 8. Consider a circuit consisting of two consecutive NOT gates, the entire circuit belongs to a CMOS 5 Volt series, if certa ...

... 6. the diagram above shows the general implementation of ________ form * POS 7. The expression _________ is an example of Commutative Law for Multiplication. * AB=BA (Page 72) 8. Consider a circuit consisting of two consecutive NOT gates, the entire circuit belongs to a CMOS 5 Volt series, if certa ...

EECS 203-1 – Winter 2002 Definitions review sheet

... it is true for all possible assignments of truth values to its variables. A contradictory expression is false for all assignments of truth values to its variables. A satisfiable formula is an expression which is true for at least one assignment. • Logical equivalence and implication in propositional ...

... it is true for all possible assignments of truth values to its variables. A contradictory expression is false for all assignments of truth values to its variables. A satisfiable formula is an expression which is true for at least one assignment. • Logical equivalence and implication in propositional ...

Lecture Notes 2

... to contradiction”. To prove a proposition x it’s suﬃcient to assume the negation ¬x of x and deduce from ¬x two contradictory statements, say, y and ¬y. The the above tautology implies that x is true. Many mathematical theorems can be formally described by an implication x → y. Call this implication ...

... to contradiction”. To prove a proposition x it’s suﬃcient to assume the negation ¬x of x and deduce from ¬x two contradictory statements, say, y and ¬y. The the above tautology implies that x is true. Many mathematical theorems can be formally described by an implication x → y. Call this implication ...

21. Solve by Elimination

... Combine the solved x value and the solved y value into an ordered pair to determine the solution for the system. ____________________. Example 2: ...

... Combine the solved x value and the solved y value into an ordered pair to determine the solution for the system. ____________________. Example 2: ...

3.1 Review 3.2 The truth table method

... • Drop every clause containing p (why?: Ci evals to true, we don’t need it anymore) • Drop every ¬p from the remaining clauses (why?: if Cj evals to true, then it doesn’t have to do with ¬p, which is false) • The formula doesn’t contain p anymore Repeat splitting the remaining variables in the remai ...

... • Drop every clause containing p (why?: Ci evals to true, we don’t need it anymore) • Drop every ¬p from the remaining clauses (why?: if Cj evals to true, then it doesn’t have to do with ¬p, which is false) • The formula doesn’t contain p anymore Repeat splitting the remaining variables in the remai ...

Normal Forms

... Proof First construct an equivalent rectified formula. Then pull the quantifiers to the front using the following equivalences from left to right as long as possible: ¬∀x F ...

... Proof First construct an equivalent rectified formula. Then pull the quantifiers to the front using the following equivalences from left to right as long as possible: ¬∀x F ...

CSE596, Fall 2015 Problem Set 1 Due Wed. Sept. 16

... (1) Design a deterministic finite automaton with alphabet {0, 1} to recognize the language of binary numbers (in standard binary notation with leading zeroes allowed) that are multiples of 5. It is your choice whether you prefer to include the empty string λ in this language, or not—you must state y ...

... (1) Design a deterministic finite automaton with alphabet {0, 1} to recognize the language of binary numbers (in standard binary notation with leading zeroes allowed) that are multiples of 5. It is your choice whether you prefer to include the empty string λ in this language, or not—you must state y ...

Homework 3

... Problem 4: Consider a set K of numbers having all eight factors of 30, that is, {1, 2, 3, 5, 6, 10, 15, 30}. Let two binary operations be LCM (least common multiple) denoted as + and GCF (greatest common factor) denoted as · . Show that: (a) The identity element for + (LCM) is 1 and that for · (GCF) ...

... Problem 4: Consider a set K of numbers having all eight factors of 30, that is, {1, 2, 3, 5, 6, 10, 15, 30}. Let two binary operations be LCM (least common multiple) denoted as + and GCF (greatest common factor) denoted as · . Show that: (a) The identity element for + (LCM) is 1 and that for · (GCF) ...

functional model

... – The rows represent inputs and the corresponding outputs. – Columns are input and output variables. • It is the simplest way to represent a circuit. • In Boolean arithmetic, the possible values are 0 and 1. So for an ninput gate, number of possible outcomes = 2n • The inputs are given in an increme ...

... – The rows represent inputs and the corresponding outputs. – Columns are input and output variables. • It is the simplest way to represent a circuit. • In Boolean arithmetic, the possible values are 0 and 1. So for an ninput gate, number of possible outcomes = 2n • The inputs are given in an increme ...

notes 25 Algebra Variables and Expressions

... • ADV Practice pg.453 #1-15 ALL • REG Practice pg.453 #2-14 EVEN https://www.khanacademy.org/math/cc-sixth-grade-math/cc-6thexpressions-and-variables/cc-6th-substitution/v/variables-andexpressions-1 https://www.khanacademy.org/math/cc-sixth-grade-math/cc-6thexpressions-and-variables/cc-6th-substitut ...

... • ADV Practice pg.453 #1-15 ALL • REG Practice pg.453 #2-14 EVEN https://www.khanacademy.org/math/cc-sixth-grade-math/cc-6thexpressions-and-variables/cc-6th-substitution/v/variables-andexpressions-1 https://www.khanacademy.org/math/cc-sixth-grade-math/cc-6thexpressions-and-variables/cc-6th-substitut ...

BOOLEAN ALGEBRA AND LOGIC SIMPLIFICATION

... • When two or more sum terms are multiplied by Boolean multiplication, the result is a • Product-of-Sum or POS expression. • • (A + B)(A + B + C) • • (A + B + C)(C + D + E)(B + C + D) • • (A + B)(A + B + C)(A + C) • The Domain of a POS expression is the set of variables contained in the expression, ...

... • When two or more sum terms are multiplied by Boolean multiplication, the result is a • Product-of-Sum or POS expression. • • (A + B)(A + B + C) • • (A + B + C)(C + D + E)(B + C + D) • • (A + B)(A + B + C)(A + C) • The Domain of a POS expression is the set of variables contained in the expression, ...

One and Two digit Addition and Subtraction - Perfect Math

... • A polynomial has multiple variables that we cannot add together, such as x and y, or x and x2 • We can actually have as many variables and exponents of variables as we can fit. • There are ways to add, subtract, multiply, and divide these still even if the variables do not add up. ...

... • A polynomial has multiple variables that we cannot add together, such as x and y, or x and x2 • We can actually have as many variables and exponents of variables as we can fit. • There are ways to add, subtract, multiply, and divide these still even if the variables do not add up. ...

Exam 1

... a) Computers are commercially successful strictly due to their speed / cost b) Placing an entire processor on a single chip made computers much cheaper c) The number of transistors on a chip is twice as many now as there were 10 years ago d) The goal of the computer architect is to make it easy to p ...

... a) Computers are commercially successful strictly due to their speed / cost b) Placing an entire processor on a single chip made computers much cheaper c) The number of transistors on a chip is twice as many now as there were 10 years ago d) The goal of the computer architect is to make it easy to p ...