# What is Truth table: Definition and 71 Discussions

A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional arguments, that is, for each combination of values taken by their logical variables. In particular, truth tables can be used to show whether a propositional expression is true for all legitimate input values, that is, logically valid.
A truth table has one column for each input variable (for example, P and Q), and one final column showing all of the possible results of the logical operation that the table represents (for example, P XOR Q). Each row of the truth table contains one possible configuration of the input variables (for instance, P=true Q=false), and the result of the operation for those values. See the examples below for further clarification. Ludwig Wittgenstein is generally credited with inventing and popularizing the truth table in his Tractatus Logico-Philosophicus, which was completed in 1918 and published in 1921. Such a system was also independently proposed in 1921 by Emil Leon Post. An even earlier iteration of the truth table has also been found in unpublished manuscripts by Charles Sanders Peirce from 1893, antedating both publications by nearly 30 years.

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1. ### B Vacuously true statements and why false implies truth

We say that an implication p --> q is vaccuously true if p is false. Since now it's impossible to have p true and q false. That is we can't check anymore whether the contrary, p being true and q being false,can be.Since p being true is non-existent. So we take the implication as true. For eg...

3. ### MHB A function given by a logical expression write in the truth table.

Hi I need help I'm lost Assignment: A function given by a logical expression Y = A.B.D + A.not (C) .D + A.not (B) .C.D + A.D write in the truth table.
4. ### Engineering 2-bit Bin Full Adder Truth Table Derivation

Hi all, I am having some troubles deriving the truth table of the following: I don't understand how does one gets to the highlight parts. For example, 10 10, when the carry is low, shouldn't the output be 0 0 1? I fully understand how to add if it is a single bit, but now with 2 bits, it is a...
5. ### MHB Construct circuit that implements truth table

Hello! (Wave) I want to construct a circuit that implements the following truth table:\begin{equation*} \begin{array}{c|c|c|c|c} P(Input) & Q(Input) & K(Output) & A(Output) \\ \hline 1 & 1 & 1 & 0 \\ 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{array} \end{equation*}In...
6. ### Engineering Circuit Analysis Question (Truth Table, Logic etc.)

Hi all, Having some problems digesting on electric circuits. Below is an example of a question and I would like to ask how do I go ahead in solving this. Firstly, for these types of questions: I have understood how to write a function table, and it goes something like this: Now what I am...
7. ### B Help on this truth table- Is this statement valid

This isn't a homework problem, but I was going to post it in there as my question is similar to one that would be asked there. However there doesn't seem to be a logic section for it so I posted it here. The question is to make a logic table to determine if the following statement is valid...
8. ### MHB Truth Table Precedence: Evaluating Implication Rules

Hello! The question is attached. I know that " $\implies$ " (implies) has precedence from right to left. But because " l- " appears after P$\implies ($Q $\implies$ R ), in my truth table do I evaluate:(P$\implies ($Q $\implies$ R ) ) $\implies$ ((P$\implies$Q ) $\implies$ R ) ) or...
9. ### I Truth, lie and random confusion

So I am very, very new to logic based questions, and in the past have solved some with relative ease but whilest scrolling through the next to find some example stuff I came across a website that gives a question and hints to the question if stuck, so I thought this would be good practice. But...
10. ### Exploring the Concept of Dual: Truth Table Homework

Homework Statement What is dual? Homework EquationsThe Attempt at a Solution Not on net. Meaning of dual i don't know and can't find.
11. ### Truth Table Homework: Equations and Solutions for T and F

Homework Statement See both images please Homework Equations It is truth table. It has T and F . The Attempt at a Solution What is that double line arrow thing? I can't find on net.
12. ### MHB Truth Table for P(x) & R(x), ~Q(x) & P(x) in U

Consider the following open propositions over the universe U = {− 4,−2, 0, 1, 3, 5, 6, 8, 10} P(x): x ≥ 4 Q(x): x 2 = 25 R(x): s is a multiple of 2 Find on a single truth table the truth-values of the following. i. P(x ) ∧ R (x ) ii. [~ Q(x )] ∧ P(x )]
13. ### Using Wolfram Alpha to Create Truth Tables for Logical Statements

Homework Statement [/B] "Use the truth table method in Boole to determine whether the conclusion is a tautological consequence of the premises." (Tet(a) ^ Small(a)) v Small(b) ------ Small(a) v Small(b) Homework Equations [/B] Taller(claire,max) v Taller(max,claire) Taller(claire,max)...
14. ### How can a NAND gate be used as a NOT gate?

Homework Statement (a) The diagram represents a NAND gate with two inputs, T1 and T2, and an output X. Copy the truth table above and complete it. Show how a NAND gate can become a NOT gate. (b) Draw up a truth table for the combination of NAND gates shown above. 2. The attempt at a...
15. ### Construct a truth table for the combination of logic gates

Homework Statement Construct a truth table for the combination of logic gates shown above. 2. The attempt at a solution First of all the combination of logic gates with added letters to the combination: The truth table: I think it should be correct. But just want to be sure.
16. ### Is Person A a Knight or a Knave on the Island?

Homework Statement A person can either be a knight (always tells the truth) or a knave (always tells a lie). On an island with three persons (A, B and C), A tells "If I am a knight, then at least one of us is a knave". Homework Equations Truth tables, logic rules. The Attempt at a Solution...
17. ### Truth table help.... did I do it correctly

Homework Statement The question I'm asking about is number two. Should've made it clearer. Homework Equations N/A The Attempt at a Solution My proof table, I'm not sure but it seems that PΞQ is not true.
18. ### MHB How Do Truth Tables and Venn Diagrams Verify Logical Equivalence?

How would I use a truth table to show that the statement p v (q ^ r) is equivalent to (p v q) ^ (p v r) or design a venn diagram for this. and show that this statement is not equivalent to (p v q) ^ r.
19. ### Full Adder Truth Table: Solving the Mystery

I don't understand the second row of the truth table. If A is 0 and B is also 0 how is there a carry of 1 and a sum of 1. 0+0=0
20. ### What Is the Correct Truth Table for This Xor Equation?

(( X xor Y ) xor Z ) xor T I hope the truth table is correct. I'm not sure, because of the T. T is always 1, right? X Y Z - (( X xor Y ) xor Z ) xor T 111 - 0 110 - 1 101 - 1 100 - 0 011 - 1 010 - 0 001 -...
21. ### Need some clarification on reading this truth table

Homework Statement Homework Equations n/a The Attempt at a Solution I understand how to read the truth table. I understand that A, B, C, D are the inputs where 10 possible inputs are shown. However, with little to no sources available, I am lost on behalf of how this image came about to...
22. ### Create Truth Table for Controller Outputs: S1, S2, R1, R2

Homework Statement The question is mostly in the picture, it has tables associated with it so I couldn't put it here.QUESTION: Make a truth table of the controller outputs enable/direction, as a function of the inputs S1, S2, R1, and R2. Since you have not been given the explicit output for...
23. ### What Is the Correct Symbolic Form and Truth Table for ~(p|q)?

<< Mentor Note -- Moved from the technical math forums, so no HH Template is shown >> I am currently working through a Finite math book Intro to finite math: second Edition Kemeny, Snell, and Thompson. One of the exercises wants me to construct a truth table for the following: ~(p|q)...
24. ### Engineering Building a logic circuit from a truth table

P.S I know I am not supposed to use attachment but there are no way for me to input the table in a clean way... same for the logic circuit drawing. Would anyone mind helping me? 1. Homework Statement Homework Equations AND gates require both input to be True to get a true ouput NOT gates...
25. ### Proving a tautology using truth table

Homework Statement ((p_r)^(q_r))<-->((p^q)_r) Prove whether the following statement is a tautology , contigency or contradiction using a truth table. Homework EquationsThe Attempt at a Solution I did the truth table, but this whole thing is one statement no? What do I compare? the first half...
26. ### Truth table to equation question (comp Sci)

Homework Statement http://imgur.com/BSntip0The Attempt at a Solution for part a, i got the solution to be F1 = xy'+x'y= x⊕y (XOR) , F2 = x .y (AND) for part b i don't understand how i would derive an equal equation that uses only NAND which is (xy') and inverter which is x'
27. ### MHB Create a Truth Table: (p ^ q) ->(p ▼ ~q)

(p ^ q) ->(p ▼ ~q) Need help creating a truth table (6 columns) for the above.Thanks
28. ### MHB Fill out the Truth Table for the following specification.

Fill out the truthe table for the following specification. Given a system with 3 inputs(y2,y1,y0) and a single output (f), where Y = y2,y1,y0 represents a 3 bit unsigned integer (Y is the decimal equivalent), determine the truth table for f such that f = 1 if and only if 1 < Y <= 6 (Y is greater...
29. ### MHB Analysis. Fill in the F column values in the truth table for the circuit.

Fill in the F column values in the truth table for the circuit. Need someone to check my work. My Answer: x y z | f | x! and y! | x! XOR z| 0 0 0| 0 | 1 | 1 0 0 1| 1 | 1 | 0 0 1 0| 1 | 0 | 1 0 1 1| 0 | 0 | 0 1 0 0| 0 | 0 | 0 1 0 1| 1 |...
30. ### Draw a truth table for x XOR 1 = x

Homework Statement Using boolean algebra, verify if the identity is true. Draw truth table to verify answers. x XOR 1 = x The Attempt at a Solution x \oplus 1 = x (x’ * 1) + (x * 1’) (x’ * 1) + (x * 0) (x’ * 1) + 0 (x’ * 1) x’ I've proven that the equation is false. How...
31. ### MHB Truth Table, Canonical Sum and Minimizing the Canonical Sum

Also would someone mind checking my work on these problems too? My answers are in BOLD 2a)Draw the truth table corresponding to $f$(X,Y,Z) = \piM(2,4,6) ANSWER: x y z | f 0 0 0| 1 0 0 1| 1 0 1 0| 0 0 1 1| 1 1 0 0| 0 1 0 1| 1 1 1 0| 0 1 1 1| 1 2b) Write out the canonical product of sums...
32. ### Understanding the truth table of → (implies)

The truth table of it is as follows for reference: p q p→q T T T T F F F T T F F T I was wondering anyone can shed some light on an easier way to memorize or think about this then just memorizing the truth table. This seems to be the least...
33. ### MHB Truth Table in Discrete Mathematics

Use a truth table to determine that "division into cases" rule of inference is valid.
34. ### Truth table, implication and equivalence

Hello, I have some questions about the truth tables for impliocation and equivalence. for implication we have: p | q | p=> q T | T | T T | F | F F | T | T F | F | T Here I do not understand the last two lines, how can we say that p implies q when...
35. ### Canonical expression and truth table

Hey, What are the steps to follow to draw a truth table and create canonical expression from a given expression A? Lets say I have A= B'.(C'+D) 1) first I have to draw the truth table for the expression above, but How could I do that since I only have one expression? 2)Then I...
36. ### Truth Table Rules of Inference

Homework Statement Use the rules of inference to prove the following: (¬p ^ q) ^ (r → p) ^ (¬r → s) ^ (s → t) ) ⇔ t. Homework Equations Rules of Inference I guess. The Attempt at a Solution Honestly I don't know where to start using the rules of inference. I drew a truth table and...
37. ### Truth table to boolean expression

I have a 9 input to 9 output truth table I constructed using Python. I am trying to find a program that can deduce the (hopefully minimized) Boolean expression for my outputs. I tried Logisim already, but when I try to paste the enormous data (512 rows), my Logisim crashes. Can anyone recommend...
38. ### Need help with truth table for P->Q and it's inverse

Hi! I'm struggling with the bellow truth tables: P = I live in Paris Q = I live in France A B C P → Q Q → P ¬ Q → ¬ P S S S S S S F S F S F F F S S S F F F S S S F F F S S F S F F S F S S S Table A and C I'm...
39. ### Engineering Boolean and truth table for circuit

I attached the problem statement. The problems I have done in the past always involved gates (AND, OR, etc...) Now this involves transistors, I honestly don't know how to go about doing this. One of the hints for the problem is to assume the ideal transistors are operating in cutoff or...
40. ### Designing a 2-Bit Half Adder: How to Add Two 2-Bit Numbers Without Carry?

I had a cpre exam last night and this was the last question. Can some one tell me the answer, I'm curious if I got it right or not. The part that tripped me up was a 2 bit "half adder". I assumed I wasn't supposed to use a full adder. So I'm not sure what to do with the carry from the first...
41. ### Diccrete Math problem: finding a proposition given a specific truth table

I've been working at this problem for a while and it seems that there should be an easier more systematic way of solving it. Here it is: Find a proposition using only p, q, ¬ and the connective ∧ with the given truth table. p q ? T T F T F F F T T F F T I know of a...
42. ### Diccrete Math problem: finding a proposition given a specific truth table

Homework Statement Find a proposition using only p, q, ¬ and the connective ∧ with the given truth table. p q ? T T F T F F F T T F F T Homework Equations The Attempt at a Solution There is a systematic way of solving these but this system requires the use of an...
43. ### Why is this truth table incorrect?

Homework Statement Create a truth table for ((P->Q) == (Q->P))Homework EquationsThe Attempt at a Solution http://www.freeimagehosting.net/newuploads/94ldj.pngI've had two quizes with problems just like this marked wrong in my college logic course. I confronted the professor about it and he...
44. ### Truth Table for A unless B , or A , only if B ?

Truth Table for "A unless B", or "A , only if B"? Hi, All: My friend is preparing for the LSAT's, and he asked me to help her with the Logic Games part. Two of the binary connectives often used in those games are 'unless', and 'only if'. I guess I'm being kind of lazy , hoping...
45. ### I am trying to constuct a truth table from Z=(A+(not)B).(B+C).((not)A+B+D)?

Homework Statement Construct a truth table to show: Z=(A+(not)B).(B+C).((not)A+B+D)Homework Equations The Attempt at a Solution from 0,0,0,0 to 1,1,1,1 i get 0,0,0,0,0,1,1,1,1,1,1,1,0,0,1,1
46. ### Find a statement S, based on P, Q, and R, that is represented by the truth table

Firstly, I apologize if this is the wrong section to post, not sure where in homework help for logic is suppose to be post. Otherwise, I have no idea how to approach this problem, I just make an equation and hope and pray it matches the results of S, or try and keep adding statements to bring...
47. ### Boolean XOR truth table for w,x,y,z

Homework Statement Recall that the Boolean addition is defined by the rule x XOR y = xy' OR x'y. Consider the Boolean function in four variables w, x, y, z defined by the formula f = w XOR y XOR xz XOR wxz XOR wyz XOR xyz XOR wxyz. A Karnaugh map method must be used to find all simple...
48. ### Help, find a statement using a truth table

i don't understand how i can find a statement in given truth table. p | q | r | S | ========== T | T | T | F| T |T | F | T | T | F | T | T | T | F | F | F | F | T | T | T | F | T | F | F | F | F | T | T | F | F | F | F | please help me.
49. ### Solving Formula for Truth Table: P & Q

I need to find the formula involving \neg, \wedge or \vee for the following truth table with the variables P and Q: P Q formula F F F F T T T F T T T F The closest I've gotten is something like P\vee(\negP\wedgeQ) which clearly doesn't work for the last row. Any ideas?
50. ### Equating Truth Table Expressions: A NAND Solution

Wasn't sure of where the most appropriate place would be for this post. If you have a truth table, say for inclusive OR, then you get a sums-of-products expression: (A * B^c) + (A^c * B) + (A * B) From this, how could I arrive at the following: A + B = ((A + B)^c)^c = (A^c * B^c)^c...