# Tagged Questions

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### Laws of equivalence needed to prove $\;q \leftrightarrow (¬p ∨ ¬q) ≡ (¬p ∧ q)\;?$

I'm not sure which laws should be applied and how I can tell for myself how to discern which laws I should use - any and all help is appreciated.
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### What is the Equivalent formula of $((a\to b) \to ((a \to c) \to (c \to a)))$

Need help to solving a logic. The question is to find an equivalent to the following logic. $((a\to b) \to ((a \to c) \to (c \to a)))$ Thanks in advance for help.
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### Conjuctive Normal Form

In Boolean logic, a formula is in conjunctive normal form or clausal normal form if it is a conjunction of clauses, where a clause is a disjunction of literals; otherwise put, it is an AND of ORs. I ...
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### How to prove this logical equivalence using different laws?

Prove that $﹁p → (q→r)$ and $q → (p∨r)$ are logically equivalent using different laws. this is my answer: $﹁p → (q→r) = q → (p∨r)$ $(q→r) = ﹁q∨r$ implication equivalence $﹁p → (q→r) = p∨(﹁q∨r)$ ...
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### About $\Sigma=\{p_2\to p_1, p_3\to p_2,\, \dots\,\}$ . . .

Suppose $$\Sigma=\{p_2\to p_1, p_3\to p_2,\, \dots\,\}.$$ Which of the following is true? Explain your answer. For any $n$, $$\Sigma\cup\{p_n, \neg p_{n+1}\}$$ is complete and ...
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### Quick Truth Table in Logic Problem

Suppose We Have: How can quickly detect how many "1" are in the truth table of above formula? (without drawing Truth Table). i think by using some inference. any idea? we know there are 11 "1"s ...
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### Discrete math and rules of inference

I recently did this rules of inference/logic question and the method I used was different from the textbook so I was wondering if my work was correct?
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### Natural deduction: $(\neg q \to\neg p)\vdash(p\to q)$ without Modus Tollens

Can anyone help me to obtain this result in natural deduction, without using modus tollens: $$(\neg q \to \neg p) \vdash ( p \to q)$$
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### I need to show the validity of the below arguments by using a truth table

I need to show the validity of $P \rightarrow Q$ $P \rightarrow R$ $\therefore P \rightarrow (R \wedge Q)$ Can i just show the truth table for $P \rightarrow Q$ and the truth table for ...
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### Discrete math proof issue

This is a question from my discrete math quiz. I was asked to prove there exists a Q(x). I used Disjunctive Syllogism to prove it. I was marked incorrectly because I used two different variables in ...
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### Proof methods Discrete Math

Claim: for any integers m and n, if $7m + 5n = 147,\,$ then $m$ is odd or $n$ is odd a) state the converse of the claim b) state the contrapositive of the claim. c) use proof by contrapositive to ...
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### Proving that $\{\to, \lnot\}$ is logically complete

Prove that $\{\to,\lnot\}$ is logically complete. It is known that $\{\land, \lor, \lnot\}$ is logically complete.
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### Rules of inference proofs

I have the following: Premise: {$p \lor q, q\rightarrow r,p \land s \rightarrow t, \lnot r, \lnot q \rightarrow u \land s$}, conclusion: $t$ I'm having a real hard understanding how to prove the ...
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### Translation of sentence into logical proposition

Given the sentence: "You can access the internet only if you are a computer science major or you are not a freshman" and functions: a = you can access the internet b = you are a computer science ...
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### Can mathematical induction be used to disprove something?

I saw this to be the rule of inference for mathematical induction : Now consider : as L.H.S. and as R.H.S.. Now if suppose, while trying to prove P(k) -> P(k+1), in the left hand side of ...
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### Proving/Disproving Product of two irrational number is irrational

I saw this question where I had to prove/disprove that: Ques. Product of two irrational number is irrational. I tried 'Proof by Contraposition'. Product of two irrational number is irrational. p ...
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### Propositional logic problem

Show that [ (p ∨ q) ∧ (p → r) ∧ (q → r) ] → r is a tautology (without a truth table). I am new to this, so I am not quite sure of how some rules can be used. Here is what I have so far: ...
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### A Proof relating to the Disjunctive normal form

Here is a problem from Kenneth Rosen's Discrete Mathematics and its Applications, Section 1.3 Suppose that a truth table in $n$ propositional variables is specified. Show that a compound ...
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### “Rules of inference” when the last premise is a conditional?

Another very basic Discrete Mathematics homework problem. I don't want the answer as much as I want to understand the question: Problem 7 For each of the following sets of premises, ...
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### Writing an expression using logic

Write an expression using letters $\land, \lor, and$ $\neg$ which has the following truth table: \begin{array}{ccc|c} P&Q&R&???\\ \hline T&T&T&F\\ T&T&F&T\\ ...
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### Are these propositions equivalent?

Statement 1: Maria will find job if she learns mathematics. Statement 2: Maria will find a job unless she does not learn mathematics. I know the answer is probably that these are same, but ...
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### Prove logical equivalence

\begin{gather} (p \to q) \equiv (\lnot p \lor q) \\ \lnot(p \land q) \equiv (\lnot p \lor \lnot q) \end{gather} Can these be proven without truth tables?
How do I prove $(\lnot p \rightarrow F)\rightarrow (p=T)\;$ using a truth table? (This tautology symbolizes a "proof by contradiction". If p being false leads to a contradiction, then p is true.)