Truth Table

4 Key excerpts on "Truth Table"

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  eBook - ePub

  Electronic Logic Circuits

  • J. Gibson(Author)
  • 2013(Publication Date)
  • Routledge

    (Publisher)

  Table 2.1 is one example for a circuit with three inputs. This table becomes the Truth Table when the state of each output is added to the table alongside every input combination. In some cases the specification will impose constraints which influence the form of the Truth Table to a great extent; in other cases the designer will have a large element of choice in forming the table.

  Example 3.1

  Devise a Truth Table for a circuit with the following behaviour. Three simple two-position switches are connected to supply inputs to a logic circuit which has a single output. The output controls a lamp and the circuit operation is such that changing the position of any switch changes the condition of the lamp (i.e. if the lamp was on it goes off and vice versa.)

  Solution

  The specification given is not complete; also the assumption is made that it is possible to design a circuit to perform as required, and this assumption may not be correct. The number of circuit inputs and outputs is defined by the specification but many other features are omitted and must be selected by the designer. This selection produces a revised specification which will allow a complete Truth Table to be devised for the circuit.

  a)  Inputs. The original specification does not state which switch position corresponds to the logic state 0, and which position corresponds to state 1. Arbitrarily choose one position, call it down, and let it represent an input of 0; the other position, up, then represents an input of 1. Also distinguish the three inputs by giving them the labels A, B and C.

  b)  Output. Again a free choice of logic states exists, it would seem sensible to choose an output of 1 to cause the lamp to be on and an output of 0 to correspond to the lamp off.

  c)  Starting position. Although the specification states how the circuit must change when any input changes it does not indicate the output for any input condition. In this example it is necessary to choose the output for a single set of inputs. A reasonable choice is to decide that if all the inputs are 0 then the output is also 0.

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  Mathematics for Information Technology

  • Alfred Basta, Stephan DeLong, Nadine Basta, , Alfred Basta, Stephan DeLong, Nadine Basta(Authors)
  • 2013(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  30. Apple Corporation manufactures computers. 31. The number of computer users in the United States is doubling every year. 32. China is the largest economy in the world. 33. The Chinese have developed the fastest supercomputer in the world. 34. Australia is the world’s largest continent. Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Logic 45 2.2 T RUTH T ABLES FOR N EGATION , C ONJUNCTION , AND D ISJUNCTION A Truth Table is a device we can use to investigate a statement containing a finite number of simple statements in combination. The table contains all possible truth-values of the statements involved and allows us to examine the conse-quences of the combinations of the various alternatives. That sounds terribly abstract, we know, but you will find that Truth Tables are actually nice con-structions with which to work, and an example or two will clarify the situation greatly. Be assured, by the way, that the notion of Truth Tables will recur later in this text when we attempt to optimize circuit pathways. First, let’s start with a peculiar fact: the number of rows of a Truth Table will always be a power of two. That is, the number of rows a Truth Table must have is 2, 4, 8, 16, 32, and so on. Why would this be the case? Every statement has two possible truth-values of its own, T or F. Whenever one simple statement is combined with another through the use of one connective, there are four pos-sible combinations to consider: listing the first statement’s truth-value first, the combinations are TT, TF, FT, and FF.

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  N4 Fault Finding and Protective Devices

  • Raimund Swart(Author)
  • 2024(Publication Date)
  • Future Managers

    (Publisher)

  Because every variable has two possible values, a number of “n” variables has 2n possible combinations of producing an outcome. The Truth Table therefore has 2n rows, one row for every possible outcome. A Truth Table can be used for a number of applications, such as: • Showing the number of outcomes of a logic function • Proving if a certain Boolean functions is equal to another • Helping to derive a Boolean function for certain outcomes • Helping to simplify complex Boolean expressions • Helping to change functions from sum-of-product form to product-of-sum form and vice versa The Truth Table has a column for every variable in the Boolean equation and a column for every operation in the equation. If a Truth Table is used to prove that Boolean expressions are equal to one another, it has a column for the function values on the left hand side of the equation and one for the function values on the right hand side. The aim is to prove that the left hand side of the equation is equal to the right hand side when these two columns match. By making use of the binary logic operations of logic “addition” and logic “multiplication”, a Truth Table can be used to simplify a logic function and to prove that the expressions provide equal results. 120 Module 3 • Binary logic and Boolean algebra Example 3.1 Make use of a Truth Table and logic “addition” and “multiplication” to prove that the following expression is true. A • (A + B) = A Solution There are two input variables, A and B. This means the Truth Table has 2 2 = 4 rows. The left hand side has two variables and two operations which requires 4 columns. The right hand side requires one column. The Truth Table therefore has five columns and four rows.

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  Computers and Microprocessors

  Made Simple

  • George H. Olsen, Ian Burdess(Authors)
  • 2016(Publication Date)
  • Made Simple

    (Publisher)

  Using these symbols to represent the operation of the circuit, a table may be drawn up which relates the output conditions due to a particular combina-tion of input conditions (Table 1). This is known as a Truth Table for the system, and lists the output states of a system as a function of the allowed input states. Table 1 therefore is the Logic Circuits and Boolean Algebra Fig. 47. Relay logic AND gate. 74 Computers and Microprocessors Truth Table for a two-input AND gate, and shows clearly that the output Y is only true when both inputs are at Logic 1 . Table 1 Inputs Output Y = A.B 1 = Closed 0 Ξ Open 1 = True 0 = False A B Y 0 0 0 0 1 0 1 0 0 1 1 1 Now consider the same set of contacts but this time, instead of connecting them in series, connect them in parallel (see Fig. 48). A connection will now be made between X and Y if either switch A OR Β is closed. Once again, it is inconvenient to describe the function of circuits of this type by using words such as OR to indicate which switches affect the output state. By replacing X O O Y Fig. 48. Relay Logic OR gate. the word OR by the symbol -f , the operation of the circuit may be des-cribed by the Boolean equation Y = A-f B. As with the AND gate previously described, the inputs A and Β and the output Y may assume one of only two states, Logic O representing an open switch and Logic 1 representing a closed switch. A Truth Table for this system may now be constructed using the same rules as before. This shows that the output is true if either one or both of the two contacts A or Β are closed. Table 2 Inputs Output 1 = Closed 0 Ξ Open 1 ΞΞ True 0 ΞΞ False Α Β Y 0 0 0 0 1 1 1 0 1 1 1 1 Finally, the only other unit to be introduced at this stage is known as an inverter. This consists basically of a single normally closed contact, which when energised will open the switch (see Fig. 49). In this system a Logic 1 applied to the switch input will open it, thereby breaking the connection

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