I am looking for some to explain what does this question want from me to do?

Determine all true value assignments, if any, for primitive statements $p, q, r, s, t$ that make each of the following compound statements false. Note: Do not answer this question by drawing truth tables. Study the compound statements and THINK about the truth values of the primitive statements. Your answers should be in English sentences.

a. $(p \land q) \land r \implies s \lor t$

b. $p \land (q \land r) \implies s \oplus t$ (where $\oplus$ means exclusive or)

Please note that I do not want the answer I want some one explains for me what does this question mean and what does it want from me?

This is my first math course. So consider that in mind!

  • $\begingroup$ We could cheat and compute the truth table, but that would be lengthy. Part of the point is that for some sentences, you can, by thinking, determine quickly the limited set of circumstances under which the sentence is false. $\endgroup$ – André Nicolas Jan 29 '12 at 22:30

Let me illustrate what’s wanted by working a similar problem. Suppose that the same question were asked about the compound statement $(p\land q)\to (s\lor t)$. An implication $A\to B$ is false if and only if $A$ is true and $B$ is false, so you want to ask yourself under what conditions on $p,q,s$, and $t$ will $p\land q$ be true and $s\lor t$ false. In words, $p\land q$ is true precisely when $p$ and $q$ are both true, and $s\lor t$ is false precisely when $s$ and $t$ are both false. We conclude, therefore, that

$(p\land q)\to (s\lor t)$ is false precisely when $p$ and $q$ are both true and $s$ and $t$ are both false.

As I read the question, this is the kind of answer that is wanted. Does that help?

  • $\begingroup$ Do that mean that the question means "when do they become false?" (a and b) $\endgroup$ – malhobayyeb Jan 29 '12 at 21:54
  • $\begingroup$ @MIH1406: Yes. Under what circumstances to those implications become false. And your answer should be an English sentence that describes the truth values of $p,q,r,s$, and $t$ that make them false. $\endgroup$ – Brian M. Scott Jan 29 '12 at 21:58

Let's understand this with an example:

If $p,q,r,s,t$ are True then (a) is

(True and True and True)$\to$ (True or True)

which simplifies to

True $\to$ True

which is a true statement. (b), on the other hand evaluates to

(True and True and True)$\to$ (True xor True)

which simplifies to

True $\to$ False

which is a false statement (rememeber the truth table of $\to$).

You are asked to find all the values of $p,q,r,s,t$ such that (a) evaluates to True.


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