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I have an assignment where I have to prove the validity of a statement, but I am not sure about what I am doing.

This is the assignment:

Is the statement $(A \wedge B \wedge C) \to D$ a valid argument? Justify your answer with a mathematical proof. Hint 1: a statement is valid if and only if its negation is a contraddiction. Hint 2: try to simplify the body of the implication (i.e., the part $A \wedge B \wedge C$) before simplifying the entire formula. Hint 3: a quantified formula is a contraddiction if and only if it is not possible to find a universe satisfying it.

I have done it like this:

I know that

$A \land B \land C \rightarrow D$

is equivalent to

$\neg(A \land B \land C) \lor D $

which is equivalent to

$\neg(A \land B \land C \land \neg D)$

Now if I replace $A \land B \land C$ with P with obtain:

$\neg (P \land \neg D)$

Know I state that:

$\forall x (x \in (A \land B \land C \rightarrow D))$

Then I negate it and see if it's a contradiction of the expression above:

$\neg(\forall x (x \in (A \land B \land C \rightarrow D)))$

Is the same thing as saying exists at least 1 that does not respect the statement:

$\exists x \neg(x \in (A \land B \land C \rightarrow D))$

And I replace $A \land B \land C$ with P with obtain:

$\exists x \neg(x \in (P \rightarrow D))$

$\exists x \neg(x \in (\neg P \lor D))$

$\exists x \neg(x \in \neg (P \land \neg D))$

Which is a contradiction of the above statement, right?

  • There are $16$ possible options, out of which, this statement is true in $15$ of them and false in the remaining one. In other words, it depends on the values of $A$, $B$, $C$ and $D$, which you have not specified (making it impossible to give a definite answer of true or false). – barak manos Oct 24 '14 at 13:20
  • Read my updated comment above. It is impossible to determine whether or not it is correct, without specifying the values of $A,B,C,D$. – barak manos Oct 24 '14 at 13:23
  • Just give a counterexample for this statement being always true (see my answer below). – barak manos Oct 24 '14 at 13:36

2 Answers2

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The statement $A \wedge B \wedge C \rightarrow D$ is not necessarily true:

$[A=true]\wedge[B=true]\wedge[C=true]\wedge[D=false]$

barak manos
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Is the statement $(A∧B∧C)→D$ a valid argument?

The statement is just one statement. It is not an argument.

An argument is a series of statements (called premises) intended to prove the truth of another statement (called the conclusion), through some accepted form of logical reasoning.

Graham Kemp
  • 129,094