I'm doing homework for this math class i'm taking, and the question is:

  1. A set of propositions is consistent if there is an assignment of truth values to each of the variables in the propositions that makes each proposition true. Is the following set of propositions consistent?

    • The system is in multi-user state if and only if it is operating normally.

    • If the system is operating normally, the kernel is functioning.

    • The kernel is not functioning or the system is in interrupt mode.

    • If the system is not in multiuser state, then it is in interrupt mode. The system is in interrupt mode.

I made a truth table and got the results, but don't know what to do with them because i don't know what the question is asking. Can someone explain?

  • $\begingroup$ Basically, the question is asking if there is one row in your truth table in which the entire right side of the table has the value true. $\endgroup$
    – 5xum
    Sep 30, 2015 at 20:40

1 Answer 1


You can abbreviate the statements as

$M$: multi-user state

$N$: operating normally

$K$: kernel functioning

$I$: in interrupt mode

so that your statments are $$M \iff N, \quad N \implies K,\quad \neg K \vee I, \quad \neg M \implies I, \quad I.$$ You want truth values so that all these statments are true. Clearly $I$ must be true, and in this case so are the third and fourth statements. Just choose values for $M$, $N$, and $K$ that make the first two statements true. One possibility is if $M$, $N$, and $K$ all share the same truth value.

  • $\begingroup$ Yes this is the answer I got. The row where MNK were false and I was true $\endgroup$
    – Andrew Kor
    Sep 30, 2015 at 21:45

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