Learning methods for math exam

Computer Sci

I am currently studying for an exam in January, and part of the exam involves questions surrounding Propositional Calculus and Truth tables. My mentor has left (a) Past exam papers, and (b) the solutions to those exam questions on Moodle and I have been studying them for the past hour or two. The mentor mentioned on numerous occasions that the exam is pretty similar year in, year out. So I’m wondering would memorisation do the trick for getting exam questions correct, or should I definitely study the theory behind such occasions – in case a different question, with different figures comes up. Or would using a few simple methods of memorisation truth table answers whilst studying past exam papers be enough to apply those same methods to an exam question, even if it had different figures?

The exam question is as follows.

Verify each of the following equivalence laws:


FIRST: A implies B..... LOGICAL.EQ........... negation.A OR B


SECOND: negation.(A implies .... LOGICAL.EQ.......... A AND(B.implies False)

Before I do out the truth table, I have devised two personal memorisation methods for the exam. The first is to simply jot down Truth/ Truth (T/T).....Truth / False (T/F).....False / Truth (F/ T).....False / False (F/T). The second is to simply remember that True and False = False (non sequitur?). Let’s say for two propositional variables A and B, that A= It is raining, and B= there are clouds in the sky. So obviously A=T and B=F as a wff is wrong because it stands for "It is raining, therefore there are no clouds in the sky".

The use of English in the book and in lessons was used to make it easier to learn the material, so these references are just means to an end. I have the layout of (a) the first two columns of the truth table laid out below, and (b) the false wff.

TT
TF
FT
FF

T&F= F

Then you have to write out the columns appropriate the question. I have separated the columns with dots. So under Column A + B I jot down the TT/ TF/ FT/ FF into column going down. Then when it comes to the third column (A.impliesB), I use two memorisations from my lessons and the book. First of all is the T&F=F rule, and second is to simply things by assuming that A=It’s raining, and B=there are clouds in the sky. So that when it comes to the third column (A.implies .....(1)A=T and B=T = T, (2) A=T and B=F = F, (3) A=F and B=T = T, and (4) A=F and B=F = T.

I’m aware that columns 4 and 7 are fairly straight foward as they are simply the reversal of the values in columns 1 and 2. For instance, if the first line of the A column is true, then the negation of that value is false if the 4th column. Likewise, if the value in the B column is false, then the column B.impliesFALSE is true/

In column 7, the negation of A.impliesB likewise, is simply the opposite of the value in the A.impliesB column.


COLUMNS 1-8


A.....B.....A.impliesB.....¬A.....¬A v B.....¬(A.impliesB).....B.impliesFALSE.....A^(B.impliesFALSE)


I ask this question because although I can get the correct answer for this particular exam question I haven’t got a clue about things such as (a) how the values in column 5 (¬A v , and column column 8 (A ^ (B.impliesFALSE) ) came about, or why that is the correct answer.

I have simply copied the same values from one column into another. For instance, to show that A.impliesB LOGICAL.EQ...negation A v B.................or that negation. (A IMPLIES ...LOGICAL.EQ... A ^ (B.implies False).........I simply put the same values from the two columns into the latter (or the columns that are "logically equivalent").

Is this a bad way to learn math, or is it possible to get away with learning math in this fashion? It’s just that I have also been reading through the book, but find this way of getting answers to be quicker. But I am not sure if I am simply memorising an old question and not actually getting to understand the logic behind such questions, so that at exam time I might do badly if a different questions comes up.

sponsoredwalk

There is absolutely no memorization involved in a truth table apart from
two things as far as I know.

1) The pattern of P, Q and R.

Okay, lets say you have 2 statements P and Q I've noticed a pattern of
true & false that most authors use.



Notice the way this is laid out, knowing the pattern of these two means
you can derive every possible relationship by just looking at the way these
two are related.

For instance, P ⋀ Q

P | Q | (P ⋀ Q)
T | T | __T__
T | F | __F__
F | T | __F__
F | F | __F__

I hope you understand why this is the case, please let me know.

2) The second thing you need to be aware of is that in a conditional or
implicative statement P ⇒ Q there is a seriously important quirk.

P | Q | (P ⇒ Q)
T | T | __T__
T | F | __F__
F | T | __T__
F | F | __T__

When the antecedent is false we simply cannot determine the truth value
of the consequent & therefore, to err on the side of caution, we denote
the statement as true as it has not been proven false conclusively!
When P is true & Q is false then we can say that (P ⇒ Q) is in fact a
false conditional statement, if P is false to begin with we can't determine
the truth value of Q as false so we assume it's a true implication as
nothing is shown to be false by assuming this.

There are three more things I want to clarify as addenda, if you have more
than two statements P and Q, say P,Q and R, the pattern of the truth
table to memorize is as follows:



I honestly think you don't need to memorize this even, you just need to
make sure that every possible relationship is expressed in the table. It
just seems easiest to memorize this as it's pretty trivial & saves labour of
checking each relationship & thinking about what you're forgetting.

The second thing is:

Computer Sci wrote: »

although I can get the correct answer for this particular exam question I haven’t got a clue about things such as (a) how the values in column 5 (¬A v , and column column 8 (A ^ (B.impliesFALSE) ) came about, or why that is the correct answer.

I have simply copied the same values from one column into another. For instance, to show that A.impliesB LOGICAL.EQ...negation A v B.................or that negation. (A IMPLIES ...LOGICAL.EQ... A ^ (B.implies False).........I simply put the same values from the two columns into the latter (or the columns that are "logically equivalent").

I wonder do you understand why (P ⇒ Q) ⇔ [(¬Q) ⇒ (¬P)] ?

P | Q |¬P |¬Q |_(P ⇒ Q)|(¬Q) ⇒ (¬P)|
T | T |_F_|_F |___T___ |____ T ____|
T | F |_F_|_T |___F___ |____ F ____|
F | T |_T_|_F |___T___ |____ T ____|
F | F |_T_|_T |___T___ |____ T ____|

You see that they have the same truth values in their table so they
logically have the same meaning. If you've got to prove, say,
(x ∈ Q) ⇒ (x² ≠ 2) i.e. (P ⇒ Q),
you can rewrite it as:
(x² = 2) ⇒ (x ∉ Q) i.e. (¬Q) ⇒ (¬P),
and still have the same logical meaning, they are logically equivalent
because they give the same answer in their truth tables no matter what
situation you're dealing with.

The third & final thing I want to mention is the difference between a
conditional statement & an implication (implicative statement). A
conditional is simple the possible configurations of a conditional
relationship, i.e. what are the possible ways that the truth value of
a conditional statement like (P ⇒ Q) can be set up? You are accounting
for all possible situations in the conditional, if P is true & Q is true what
is the truth value of the conditional (P ⇒ Q) ? If P is false & Q is true
what is the truth value of the conditional (P ⇒ Q)? etc...
An implication is the process of assigning a truth value to the statement
(P ⇒ Q), i.e. (P ⇒ Q) must be true so we must assess the actual
meaning/content of the statements P & Q. The big difference is that
the meaning of the statements matters in an implication & you are
either showing a specific implication is true of false based on the content
of those statements.

If you have any more questions fire away, I'm no expert by any means but
I'm pretty confident the above is all correct. I better to ask you:

You understand the inclusionary nature of "and" ⋀ right?
You understand the exclusionary nature of "or" ⋁ right?
You understand why the truth table of (¬P) is the mirror opposite of P's right?
You understand that in writing truth tables the content of the statements
is essentially meaningless & that only after the table is complete & you
are looking at specific statements actual meaning you are just picking
the specific relationship that applies in that situation right?

Computer Sci wrote: »

A.impliesB LOGICAL.EQ...negation A v B

I don't think that's right, I'd have to see the way you've shown this to
be true on a truth table.

Computer Sci wrote: »

(A ^ (B.impliesFALSE)

What does (B.impliesFALSE) mean?