Thread: Coursework help

1. Coursework help

Hi, i've just started a degree in Computer Science and have a discrete math assignment to hand in tommorow. Could anyone please help?

(I'm not asking you to do it for me but explain how to do it or show me how to do 1 part in each question?)

These are the questions

1. Explain the difference between the following three offers, by analysing their logical form:

a) You can watch TV if you tidy your room
b) You can watch TV only if you tidy your room
c) You can watch TV if, and only if, you tidy your room

Which offer should a logical parent make to their children?

2. Construct the truth tables of the following compound formulas:
i) P => Q v R
ii) (P v Q) ^ (P => Q)

3. Use truth tables to show that the following formulas are always true:
i) P ^ Q <=> Q ^ P
ii) P ^ (Q v R) <=> (P ^ Q) v (P ^ R)
iii) ¬(P v Q) <=> ¬P ^ ¬Q

4. An enormous amount of loot has been stolen from a store. The criminal (or criminals) took the hesit away in the car. Three well known criminals A, B and C were brought to Scotland Yard for questioning. The following facts were ascertained:

a) No one other than A, B and C was involved in the robbery
b) C never pulls a job without using A (and possible others) as an accomplice
c) B does not know how to drive

Is A innocent or quilty?

[Hint: A way to answer this question is to rewrite the recorded facts using propositional logic, with basic statements of the form A is guilty, B is guilty, C is guilty, respectively. The next step wuld be to write down the truth tables for the formulas obtained. Keeping in mind that formulas representing facts are always true, you will then be able to deduce an answer tot he question.}

2. Originally Posted by sahil
Hi, i've just started a degree in Computer Science and have a discrete math assignment to hand in tommorow. Could anyone please help?

(I'm not asking you to do it for me but explain how to do it or show me how to do 1 part in each question?)

These are the questions

1. Explain the difference between the following three offers, by analysing their logical form:

a) You can watch TV if you tidy your room
b) You can watch TV only if you tidy your room
c) You can watch TV if, and only if, you tidy your room

Which offer should a logical parent make to their children?
Note that P => Q is true even if P is false. Also note that the implication is only false is P is true, but Q is false.

2. Construct the truth tables of the following compound formulas:
i) P => Q v R
ii) (P v Q) ^ (P => Q)
what trouble are you having doing truth tables? the first table should look like this:
$\displaystyle \displaystyle \begin{array}{|c|c|c|c|c|} \hline \bold P & \bold Q & \bold R & \bold Q \vee \bold R & \bold P \implies (\bold Q \vee \bold R) \\ \hline T & T & T & & \\ \hline T & T & F & & \\ \hline T & F & T & & \\ \hline T & F & F & & \\ \hline F & T & T & & \\ \hline F & T & F & & \\ \hline F & F & T & & \\ \hline F & F & F & & \\ \hline \end{array}$

the second should look like this:

$\displaystyle \displaystyle \begin{array}{|c|c|c|c|c|c|} \hline \bold P & \bold Q & \bold R & \bold P \vee \bold Q & \bold P \implies \bold Q & (\bold P \vee \bold Q) \wedge (\bold P \implies \bold Q) \\ \hline T & T & T & & & \\ \hline T & T & F & & & \\ \hline T & F & T & & & \\ \hline T & F & F & & & \\ \hline F & T & T & & & \\ \hline F & T & F & & & \\ \hline F & F & T & & & \\ \hline F & F & F & & & \\ \hline \end{array}$

Now fill out the tables based on the rules you know for each logical expression

3. Use truth tables to show that the following formulas are always true:
i) P ^ Q <=> Q ^ P
ii) P ^ (Q v R) <=> (P ^ Q) v (P ^ R)
iii) ¬(P v Q) <=> ¬P ^ ¬Q
construct truth tables similar to how i did in the last problem. the column quoted in each question part should have all T values in them if you fill out the tables right.

4. An enormous amount of loot has been stolen from a store. The criminal (or criminals) took the heist away in the car. Three well known criminals A, B and C were brought to Scotland Yard for questioning. The following facts were ascertained:

a) No one other than A, B and C was involved in the robbery
b) C never pulls a job without using A (and possible others) as an accomplice
c) B does not know how to drive

Is A innocent or guilty?

[Hint: A way to answer this question is to rewrite the recorded facts using propositional logic, with basic statements of the form A is guilty, B is guilty, C is guilty, respectively. The next step wuld be to write down the truth tables for the formulas obtained. Keeping in mind that formulas representing facts are always true, you will then be able to deduce an answer tot he question.}
Let P(S) be the statement "S is guilty".

Then (a) says $\displaystyle \displaystyle P(A) \vee P(B) \vee P(C)$ (is true)

(b) says $\displaystyle \displaystyle P(C) \implies P(A)$

(c) says $\displaystyle \displaystyle P(B) \implies (P(A) \vee P(C))$

That is, if B is guilty, either A or C must be guilty also, because B would need one of them to accompany him to drive.

You need to ascertain whether P(A) is always T given all these cases are true. You can do this via truth table.

3. Thank you.