Wikipedia:Reference desk/Archives/Mathematics/2011 November 8

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November 8

Z transform of {1}

Could you please explain how we obtain Z[{1}] = ${\frac {z}{-1+z}}$ by using the very definition of Z transform Thank you in advance --LijoJames (talk) 16:57, 8 November 2011 (UTC)

Assuming you mean the unilateral case and the constant sequence 1, it's a geometric series. Note that geometric series are usually stated with exponent n, while Z transform has exponent -n. So you'll need to remove the negative from the exponent, then apply the standard formula for a geometric series, then simplify the resulting expression. You'll get the desired answer.--121.74.125.249 (talk) 19:45, 8 November 2011 (UTC)

Truth table

I'm looking for an expression whose truth table is TFFTTFFT. It has three letters, its primary operator is an inclusive disjunction (V), and it may or may nor use a conjunction and some negatives as secondary operators. 132.208.165.53 (talk) 22:42, 8 November 2011 (UTC)

That's just an Xor of the first two arguments. Since the first four values (TFFT) simply repeat for the second four values, the third argument doesn't appear in the expression at all. Looie496 (talk) 22:53, 8 November 2011 (UTC)

I think it's in the form ((P & Q) V R)) or ((P V Q) V R)) with negatives in some of the variables, but I have to test the table each time to check. I kind of wish there was an online dictionary of truth tables. 132.208.165.53 (talk) 22:57, 8 November 2011 (UTC)

Please see truth table. You will not find anything like TFFTTFFT. You will find tables (rows and columns). That is why it will be difficult to find TFFTTFFT when searching for truth tables online. -- kainaw™ 23:19, 8 November 2011 (UTC)

I know all the rules on how to make truth tables, as mentioned in that page, it's just that I'm having trouble locating this one particular three-variable expression (TFFTTFFT). 132.208.165.53 (talk) 23:21, 8 November 2011 (UTC)

Looie496 is on the right track, but you really want the negation of the Xor. Note that the article he linked states

${\begin{matrix}p\oplus q&=&\lnot ((p\land q)\lor (\lnot p\land \lnot q))\end{matrix}}$ .

What happens when you negate that? -- ToE^T 00:10, 9 November 2011 (UTC)

Can I rephrase the question ? I'm looking for the disjunctive normal form of an expression with TFFTTFFT as its truth table, but it isn't really necessary to now what exactly that expression is. 76.67.166.12 (talk) 17:33, 9 November 2011 (UTC)

I am certain that I could provide a quick answer, but I cannot get past your use of "TFFTTFFT" and "truth table". I teach binary logic in many courses. I teach DNF in many courses. But, in over 20 years of doing this, I have never ever seen a truth table that is just a line of T's and F's. A truth table has the variables on one side and the expression on the other. Then, there are rows for each possible value for each variable and the result of the expression using those variables. Your "TFFTTFFT" has no variables and no expression. -- kainaw™ 17:43, 9 November 2011 (UTC)

I know what a truth table is, but I was giving the truth value of a certain table without the corresponding expression. For example, (P) has a table of T-F. (P & Q) has a table of TFFF. (P -> Q) has a value of TFTT.

What I'm looking for specifically is the complete normal forms, both the disjunctive and the conjunctive, of the operation whose truth table is TFFTTFFT. I've also been told that knowing the literal operation corresponding to TFFTTFFT is not really necessary to solving the problem, which has to do with complete normal forms.

After getting an explanation of what you mean by TFFTTFFT, this is a rather straightforward problem. First, notice that TFFT repeats twice. So, A is ignored. (I'm using A, B, and C). The first term will be (A&!A) to force it to false and ignore it. The rest is the negation of XOR. So, XOR would be (B&!C)|(!B&C). The negation is (!B|C)&(B|!C). Translating that to DNF produces (B&C)|(!B&!C). Adding back the first term, you get (A&!A)|(B&C)|(!B&!C). -- kainaw™ 18:21, 9 November 2011 (UTC)

Is it harder to find the CNF ? And are you using the exclamation mark (!) as a negative operator ? — Preceding unsigned comment added by 76.67.166.12 (talk) 18:26, 9 November 2011 (UTC)

Yes. ! means "not". Some people use ~ and some use ^ and some use -. I wasn't sure what you were using. For CNF, you want (!B|C)&(B|!C), which was in the answer above. A is, again, ignored. But, you always want it true to ignore it in CNF. So, you use (A|!A)&(!B|C)&(B|!C). -- kainaw™ 18:33, 9 November 2011 (UTC)

Okay, thanks for the help, Kainaw. 76.67.166.12 (talk) 18:34, 9 November 2011 (UTC)

Kainaw, what is your (A&!A) for? (A&!A)=0 and 0|P = P, so why bother? (Similarly with (A|!A) in the CNF?) It's not like they create a full disjunctive normal form, which requires that "each of its variables appears exactly once in every clause", as (A&B&C)|(A&!B&!C)|(!A&B&C)|(!A&!B&!C). -- ToE^T 00:51, 10 November 2011 (UTC)

I stated that A is ignored. I then used that to explain how it can be included and ignored. I could have just pretended that A didn't exist and left some people wondering where it went, but I felt it was a complete answer to state that it was ignored and show how you can include it while ignoring it. -- kainaw™ 02:00, 10 November 2011 (UTC)

Does anybody know what the proper response (DNF / CNF) would be if A wasn't automatically ignored ? This question intrigues me. 132.208.143.155 (talk) 02:34, 10 November 2011 (UTC)

For this truth table, the full DNF is (!A&!B&!C)|(!A&B&C)|(A&!B&!C)|(A&B&C) and the full CNF is (A|B|!C)&(A|!B|C)&(!A|B|!C)&(!A|!B|C). Is that what you are asking? -- ToE^T 07:13, 10 November 2011 (UTC)

The recipe for generating these forms from a set of truth values is easy and straight forward, and more importantly, it offers an insight into the meaning of the form. The full disjunctive normal form is generated by ORing together several clauses -- one clause for each true value in the truth table. Each clause is generated by ANDing together all the variables or their negations, as required to yield a true for that line of the truth table. The result is the answer to the question "what is true?" and represents the truth table one true line at a time. OP asked about a truth table with a result column of TFFTTFFT. Here, the true lines are (A,B,C) in {(0,0,0), (0,1,1), (1,0,0), (1,1,1)}. Negating the variables where they are 0 gets you these four clauses: (!A&!B&!C), (!A&B&C), (A&!B&!C), (A&B&C). OR those together and you have the full DNF. Does that make sense? Do you see the inherent relation between the truth table and the full DNF? -- ToE^T 12:17, 10 November 2011 (UTC)

OK, thanks again. I already knew the definitions of FDNF and FCNF, but I had problems connecting the dots for this particular table. By the way, I was posting from two different computers, which is why two different IP adresses show up in this discussion. 76.67.166.12 (talk) 15:54, 10 November 2011 (UTC)