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I am a beginner in logic.

With the premise that (q ∧ r) is false, how can I prove that ∼p → (q ∧ r) is invalid?

This is the last part of a logic problem. I have built this truth table:

p | q | r | ~p | (q ∧ r)
T | T | T | F  |  T
T | T | F | F  |  F
T | F | T | F  |  F
T | F | F | F  |  F
F | T | T | T  |  T
F | T | F | T  |  F
F | F | T | T  |  F
F | F | F | T  |  F
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    $\begingroup$ You can build a truth table and consider all possible values for your variables $\endgroup$
    – Vasili
    Aug 15 at 15:00
  • $\begingroup$ Do you know that $p$ is not true? $\endgroup$
    – John Douma
    Aug 15 at 15:04
  • $\begingroup$ Yes, there are (p ∨ ∼s) is True, which means p is True and ~s is True. $\endgroup$
    – tttio
    Aug 15 at 15:12
  • $\begingroup$ I do build a truth table, but I'm not sure I do a right thing, now I put my truth table on. $\endgroup$
    – tttio
    Aug 15 at 15:14
  • $\begingroup$ If $p$ is true then the statement is true because "false" implies everything. $\endgroup$
    – John Douma
    Aug 15 at 15:15

3 Answers 3

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With the premise that (q ∧ r) is false, how can I prove that ∼p → (q ∧ r) is invalid?

Since $\color\red{(q ∧ r)}$ is false, $(¬p → \color\red{(q ∧ r)})$ and $p$ have the same truth value.

Thus, since $p$ can be false, $(¬p → (q ∧ r))$ correspondingly can be false, which is what I think you're wanting to prove.

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It sometimes true (e.g. when P, Q and R are all true) and sometimes false (e.g.when P, Q and R are all false). It is a contingency. Here is the truth table.

enter image description here

Source: https://www.erpelstolz.at/gateway/TruthTable.html

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With the premise that (q ∧ r) is false, how can I prove that ∼p → (q ∧ r) is invalid?

Additionally to ryang answer, I will provide a more algebraic approach to this problem.

We use: $a \rightarrow b \equiv \neg a \lor b$

The original proposition can be written as

$\neg p \rightarrow (q \land r) \equiv p \lor (q \land r) \equiv p \lor F \equiv p$

So, $\neg p \rightarrow (q \land r)$ is true if an only if $p$ is true.

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