Home ADVANCED LEVEL TOPIC 3: LOGIC (II) ~ ADV MATHEMATICS FORM 5

TOPIC 3: LOGIC (II) ~ ADV MATHEMATICS FORM 5

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LOGIC

LOGIC (II)

SENTENCE HAVING A GIVEN TRUTH TABLE
Example.1 Find a sentence which has the following truth table
P
Q
T
T
T
T
F
T
F
T
T
F
F
F
Step: 1. Mark lines which are T in last column
2. Basic conjunction of P and Q
3. Required sentence is the disjunctions of the above basic conjunction
P
Q
Basic conjunction
T
T
T
P Q
T
F
T
P ~Q
F
T
T
~ P Q
F
F
F
Required sentence (P Q) V (P ~ Q) V (~ P Q)
Example. 2
Find a sentence having the truth table below
P
Q
R
T
T
T
T
T
T
F
F
T
F
T
F
T
F
F
T
F
T
T
F
F
T
F
T
F
F
T
F
F
F
F
F
Solution
P
Q
R
Basic conjunction
T
T
T
T
P QR
T
T
F
F
T
F
T
F
T
F
F
T
P ~ Q ~ R
F
T
T
F
F
T
F
F
F
F
T
F
F
F
F
F
Required sentence is (P Q R) V (P ~ Q ~ R)
Example. 3
Find a sentence having the following truth table and simplify it.
P
Q
T
T
T
T
F
F
F
T
T
F
F
T
SOLUTION
P
Q
Basic conjunction
T
T
T
P Q
T
F
F
F
T
T
~ P Q
F
F
T
~ P ~Q
The required sentence is (P Q) V (~ P Q) V (~ P ~ Q)
To simplify;
(P Q) V (~P Q) V (~P ~Q) = (P Q) V [~P (Q V ~Q)…..distributive law
= (P Q) V [~ P t] ……compliment law
= (P Q) V [~ P] ……..identity
= (P V ~P) (~P V Q)…….. Distributive
= t (~P V Q) ………compliment
= (~P V Q) ………identity
Note
P → Q ≡ ~ P V Q
QUESTIONS
1. for each of the following truth tables (a), (b) and (c) construct a compound sentence having that truth table.
P
Q
R
(a)
(b)
(c)
T
T
T
T
T
F
T
T
F
F
T
T
T
F
T
T
T
T
T
F
F
F
T
F
F
T
T
F
F
F
F
T
F
F
F
F
F
F
T
F
F
F
F
F
F
F
F
T
Solution
P
Q
R
(a)
(b)
(c)
Basic conjunction of (a)
Basic conjunction of (b)
Basic conjunction of ( c)
T
T
T
T
T
F
P Q R
P Q R
T
T
F
F
T
T
P Q ~ R
P Q ~ R
T
F
T
T
T
T
P ~ Q R
P ~ Q R
P ~ Q R
T
F
F
F
T
F
P ~Q ~ R
F
T
T
F
F
F
F
T
F
F
F
F
F
F
T
F
F
F
F
F
F
F
F
T
~ P ~ Q ~ R
→The required sentence for (a) is (P Q R) V (P ~ Q R)
→The required sentence for (b) is (P Q R) V (P Q ~R) V (P~Q R) V (P ~Q~ R)
→The required sentence for (c) is (P Q ~R) V (P ~Q R) V (~P ~Q ~R)
2. i) construct a truth table for ~ (P → Q)
ii) Write a compound sentence having that truth table (involving ~, , v)
3. Repeat for the following sentence
i) ~ P → ~Q ii) ~ p Lh5Faonsptdemchiqgca7Mhtkb5Hetiozlttuhs1Ymk0Yytnlkqage6Iy12 Ne6Kgmzy5Lfnhptdnluitbiuowyxvp9Ag Jrbgpp Hcucdotmun4Sswu1Hpqrrmqfwaqkv 5Odg Q
More question
1. Find a compound sentence having components P and Q which is true and only if exactly one of its components P, Q is true.
2. Find a compound sentence having components P, Q and R which is true only if exactly two of P, Q and R are true.
3. Give an example of sentence having one component which is always true
4. Give an example of a compound sentence having one component which is always false
5. Use laws of algebra of propositions to simplify ~ (p V q) (~ p q)
6. Show that p N Ms9Klsl Dvasbx5Zfmxv60L3Wqjjwwcznjvs Qe0Phb1Isn Hfnxs6Xtq2Wc6Bs793Eyzkyyz1Y9Hqwp5Qwlbiksqf2Cfnwjyo3Gi9Ma Pyezfj8Isoq4Ia08T9Q4Iiv93Zq q and ~ p v q are logically equivalent
7. If Apq Yamvbipefasvce0Ibr G9Pqemtm116Q4Wgyspnkeohoxthi2I7Lqumzey9Xx Jfbhskck3W7Fbwu1Ecc Hu30Evwwo22Qj64 p q and NpYamvbipefasvce0Ibr G9Pqemtm116Q4Wgyspnkeohoxthi2I7Lqumzey9Xx Jfbhskck3W7Fbwu1Ecc Hu30Evwwo22Qj64 ~ p write the following without ~ and A
i) ~ (p q)
ii)~ (p ~q)
iii) ~ (~ p q)
iv) ~ (p ~ q)
QUESTIONS
1. Rewrite the following without using the conditional
i) If it is cold, he wears a hat
ii) If productivity increases, then wages rise
2. Determine the truth value of the following
i) 2 + 2 = 4 if and only if 3 + 6 = 9
ii) 2 + 2 = 4 if and only if 5 + 1 = 2
iii) 1 + 1 = 2 if and only if 3 + 2 = 8
iv) 1 + 2 = 5 if and only if 3 + 1 = 4
3. Prove by truth table
i) ~ (p Lh5Faonsptdemchiqgca7Mhtkb5Hetiozlttuhs1Ymk0Yytnlkqage6Iy12 Ne6Kgmzy5Lfnhptdnluitbiuowyxvp9Ag Jrbgpp Hcucdotmun4Sswu1Hpqrrmqfwaqkv 5Odg q) ≡ p Lh5Faonsptdemchiqgca7Mhtkb5Hetiozlttuhs1Ymk0Yytnlkqage6Iy12 Ne6Kgmzy5Lfnhptdnluitbiuowyxvp9Ag Jrbgpp Hcucdotmun4Sswu1Hpqrrmqfwaqkv 5Odg ~q
ii) ~ (p Lh5Faonsptdemchiqgca7Mhtkb5Hetiozlttuhs1Ymk0Yytnlkqage6Iy12 Ne6Kgmzy5Lfnhptdnluitbiuowyxvp9Ag Jrbgpp Hcucdotmun4Sswu1Hpqrrmqfwaqkv 5Odg q) ≡ ~ p Lh5Faonsptdemchiqgca7Mhtkb5Hetiozlttuhs1Ymk0Yytnlkqage6Iy12 Ne6Kgmzy5Lfnhptdnluitbiuowyxvp9Ag Jrbgpp Hcucdotmun4Sswu1Hpqrrmqfwaqkv 5Odg q
4. Prove the conditional distributes over conjunction i.e.
[p → (q r)] ≡ (p → q) (p → r)
5. Let p denote ‘’ it is cold’’ and let q denote ” it rains “. Write the following statement in symbolic form
i) It rains only if it is cold.
ii) A necessary condition for it to be cold is that it rains.
iii) A sufficient condition for it to be cold is that it rains
iv) It never rains when it is cold.
6. a) Write the inverse of the converse of the conditional
” If a quadrilateral is a square then it is a rectangle”
b) Write the inverse of the converse of the contra positive of
“If the diagonals of the rhombus are perpendicular then it is a square”
LOGICAL IMPLICATIONS
A proposition P is said to be logically imply a proposition Q if p → Q is a tautology
Example
Show that p logically implies p v q
Solution; Construct a truth table for p → (p v q)
P
q
P v q
P → (p v q)
T
T
T
T
T
F
T
T
F
T
T
T
F
F
F
T
Since column 4 is a tautology then p logically implies p v q
ARGUMENTS
An argument in logic is a declaration that a given set of proposition p1, p2, p3….pn called premises yields to another proposition Q called a conclusion such as argument is denoted by p1, p2….pn Tuxkwgt4Ok88Pitjtvbamwkoqvkndo6Gshrknymjkwxasrjyds5Tjclkq47Erz2Nbk1Dgqqluyl4Q
Example of an argument
If I like mathematics, then I will study, either I study or I fail. But I failed therefore I do not like mathematics.
VALIDITY OF AN ARGUMENT
Validity of an argument is determined as follows
An argument P1, P2, P3… Pn Ockn 5Dgen2Aeud5Ddoek33Dl8Oznzopv3Klwfzqtsglehhaopulm0Twcwhjny9Evo 6Xot0Njafarr9By7Qzwtj75Jimfckqyei5Ascspqq9N9Lqlaj8Whhsxshexffo6 Caj0 Q is valid if Q is true whenever all the premises P1, P2, P3… Pn are true
→Validity of an argument is also determined if and only if the proposition (P1 P2 P3 ….. Pn) → Q is a tautology
Example
Prove whether the following argument is valid or not P, P → Q Ockn 5Dgen2Aeud5Ddoek33Dl8Oznzopv3Klwfzqtsglehhaopulm0Twcwhjny9Evo 6Xot0Njafarr9By7Qzwtj75Jimfckqyei5Ascspqq9N9Lqlaj8Whhsxshexffo6 Caj0 Q
Solution:
Draw a truth table for [P P → Q] → Q
P
Q
P → Q
P (p → Q)
P (p → Q) → Q
T
T
T
T
T
T
F
F
F
T
F
T
T
F
T
F
F
T
F
T
1. Since in row 1 the conclusion is true and all the premises are true then the argument is valid
2. Since column 5 is a tautology then the argument is valid
QUESTION
Use the truth table to show whether the given argument is valid or not
P → Q, Q → R Ockn 5Dgen2Aeud5Ddoek33Dl8Oznzopv3Klwfzqtsglehhaopulm0Twcwhjny9Evo 6Xot0Njafarr9By7Qzwtj75Jimfckqyei5Ascspqq9N9Lqlaj8Whhsxshexffo6 Caj0 P → R
Example
Symbolize the given argument and then test its validity
*If I like mathematics, then I will study, either I study or I fail. But I failed, therefore I do not like mathematics.
Solution.
The given argument is symbolized as follows
Let p ≡ I like mathematics
q ≡ I will study
r ≡ I fail
Then given argument is as follows
P → q, q v r, r, Ockn 5Dgen2Aeud5Ddoek33Dl8Oznzopv3Klwfzqtsglehhaopulm0Twcwhjny9Evo 6Xot0Njafarr9By7Qzwtj75Jimfckqyei5Ascspqq9N9Lqlaj8Whhsxshexffo6 Caj0~p
Testing the validity
[(p → q) (q r)r] → ~p
P
Q
r
P → q
q v r
3 4 5
~ p
6 → 7
T
T
T
T
T
T
F
F
T
T
F
T
T
F
F
T
T
F
T
F
T
F
F
T
T
F
F
F
F
F
F
T
F
T
T
T
T
T
T
T
F
T
F
T
T
F
T
T
F
F
T
T
T
T
T
T
F
F
F
T
F
F
T
T
Since column 8 is not a tautology the given argument is not valid
QUESTIONS
1. Translate the following arguments in symbolic form and then test its validity
i) If London is not in Denmark, then Paris is not in France. But Paris is in France, therefore London is in Denmark
ii) If I work I cannot study. Either I work or I pass mathematics. I passed mathematics therefore I studied.
iii) If I buy books, I lose money. I bought books, therefore I lost money
2. Determine the validity of
i) p → q, ~q Ockn 5Dgen2Aeud5Ddoek33Dl8Oznzopv3Klwfzqtsglehhaopulm0Twcwhjny9Evo 6Xot0Njafarr9By7Qzwtj75Jimfckqyei5Ascspqq9N9Lqlaj8Whhsxshexffo6 Caj0 ~p
ii)~p → q, p Ockn 5Dgen2Aeud5Ddoek33Dl8Oznzopv3Klwfzqtsglehhaopulm0Twcwhjny9Evo 6Xot0Njafarr9By7Qzwtj75Jimfckqyei5Ascspqq9N9Lqlaj8Whhsxshexffo6 Caj0 ~q
iii) [p → ~ q], r → q, r Ockn 5Dgen2Aeud5Ddoek33Dl8Oznzopv3Klwfzqtsglehhaopulm0Twcwhjny9Evo 6Xot0Njafarr9By7Qzwtj75Jimfckqyei5Ascspqq9N9Lqlaj8Whhsxshexffo6 Caj0 ~p
ELECTRICAL NETWORK
Electrical network is an arrangement of worse and switches that will accomplish a particular task e.g. lighting a lamp, turning a motor, etc
The figure below shows an electrical network
Jcuevea Uj8Ws2Spcfd3Mkcwg2Xdmhnsjvjztj7Ikhjwwz
When the switch p is closed the current flows between T1 and T2
The above network simplifies to the following network
Kcgclidtkvkvejcguvzwgwm7Tqsbi2S7 Nbmtpzjo99Dp5Wr7D211G98Ausrtuhrfx9Vlovok Qml95Dcgozjcrg6Icfmm6Uc4Xv Ge6Bwemnmbhsh Ubtwydwlukswnbeeqh30
Relationship between statement in logic and network
A Zvimjrrgn9Aarrlgwqv66Uuaxeqg4Eesjmrsm7 Ti1Jxhonxpky9Rq8Ac2Dvd Bdd5Wfxnysojx6V Ecpmhy J 2Nfovynzqnbwhlrhulzl5I63Pcp4T4Qk4 Uc0D4H1Clt0G

A SERIES AND PARALLEL CONNECTION OF SWITCHES
A series connection of switches
The following switches are connected in series
Cn1Ufftnox Bemhx5Dwiiamfa Ghga7Mw Fq3Vjdbfmtyij3Lltiekseeui9Zkfscww J Govaazlo5Wjhm8Qbza5Ffaqqedp6V2A9Sc3Opkaojvsjuetns8Hvluclozikbyrjy
The current flow between T1 and T2 when both switches are closed current flows when p Q is true
A parallel connection of switches
Czgd8Ppcjruo1Ild8G8Cjvxgw0Xhhvc0Jpai Tnu0Qglbekyvcsmiwhvbpmxtgo5Nm2E2Usn9S4Y7Kw1I0Pg Dic0Inklstyogunjqm Ggl5Kmpyvtkifvex6Vghnwesrjffpfc
The current will flow when either one of the switches is closed.
Currents flow when P V Q is true
Example
Consider the electrical network below

i) Construct a compound statement presenting the network above
ii) Find possible switch setting that will allow the current to flow between T1 and T2
Solution
Note i) current flows between T1 and T2 when switch p is closed i.e. p is true OR
ii) The current flows between T1 and T2 when switch switches q and r are closed i.e. Q R is true.
The required compound statement is p v (Q R)
iii) To find possible switch setting, draw a truth table P v (Q R)
P
Q
R
Q R
P V (Q R)
Current flows yes or No
T
T
T
T
T
Yes
T
T
F
F
T
Yes
T
F
T
F
T
Yes
T
F
F
F
T
Yes
F
T
T
T
T
Yes
F
T
F
F
F
No
F
F
T
F
F
No
F
F
F
F
F
No
Possible switch setting
P
Q
r
Closed
Closed
Closed
Closed
Closed
Closed
Closed
Closed
Closed
Closed
Closed
Closed
Closed
Closed
Closed
Questions
1. Construct compound statement that correspond to the networks
2Wajwnnikar3Jz95Nt335Mwn Rh0K4Leuoxapefibq Hme9Iofi
Solution
The current will flow when all three switches p, q, and r are closed i.e. p q r
The required compound statement is P Q R
Qrohhs5O7 Eedezzxfxlpnb0Lgohhquh6Tbkc2T9Jwi87Zddb7Xfji0Kvbf 0Yu4M02I3Qqe2Zyw9Yobphylziz9871Am3Faveftudifek6Yukepnejt8St4Quvguiti0Ecvae
The required component statement is (P q)

V3Mmkkrzl5Clnw76Simmckh4Yvlb2Kg8Ljxl3Z4Inryw6Lf8Oea64Xuzi5Goispioysmtbratwq3Cebocvpz Wwabuevg0Ve7Dlfe4Die2Ps4Wkbodpcwqxfib249Bchv7Uf5Jq

The required compound statement is (p q) V (rs)
Hajvxojpzqcvv4N1Wp0Tby Qsl1Gniris0Qtdk7Uabba7Y5Thaat8F

The required compound statement is P V Q V R
N5Kv Dgmqi2Ntojwyzdr6X5Muc6L57Rpj 0Kfcriybls80Miecettmjck4Xthe5Nsgx33 W Fit8Iapfj9Hluoyu27B3Nc96Zpfpee0Supt7Yc7Iqnt8Z8 Oghgtrcqciaeym C
The required compound statement is p (q V (r s))


Mtze3Ezowific0Crjsbndfxmjtbnqp69Ousyn72I L Su9Tzgqb7Bkhvpfol9Uvkbux5Iqij

The required compound statement is (P Q R) S
2. In electrical network of (ii) find possible switch setting that will allow the current to flow between T1 and T2
ii) (P Q) R
P
Q
R
P V Q
(P V Q) R
T
T
T
T
T
T
T
F
T
F
T
F
T
T
T
T
F
F
T
F
F
T
T
T
T
F
T
F
T
F
F
F
T
F
F
F
F
F
F
F



Possible switch settings
P
Q
R
Closed
Closed
Closed
Closed
Open
closed
From statements to network
Example
Draw a network for the statement (p v Q) (R S)
Solutions
Corresponding network is shown below

Dn0W0T3Ko98Lc5Jp8Fenqlrzkfdutbyru8Okp5Lw9Xncu1Fyuum Vhgocifkr9G4X2Ugcoxksocgc Pqo2Apdokllt W8Ib Pwduaiwyssk8H3Mgifxppdo Qnqukxes28N2E3S
Questions
Draw network for the following statements
1. [P Q (R S)]
2. [(P Q) (R V S)]
3. [P V (Q S) V (R T)]
4. (Q V(R V S) V P)
5. [P V (Q (R S)]
COMPLEX SWITCHES
These operates as follows
i) When one switch is closed, the other one closes also
ii) When one switch is closed, the other one opens
Refers to the diagram
Jxclqwdev54D1Midsttmygescwb9Dqyqfuycqefil7Kgia3Ood3Afqishnmhlffdzdrwps0Apt R8Eecqo4Eyas Cgh3Wmsjps5Qkngsdsflpq5Fwxnbsonrhxtlhtzfn K Xr0


The compound relating to flow of electrical current is given
(P Q) V [P (~ Q V R)]
To find possible switch setting that will allow the current to flow between T1 and T2
Draw a truth table for (P Q) V [ P ( ~ Q V R)]
1 2 3 4 5 6 7 8
P
Q
R
P Q
~ Q
~ Q V R
P (~ Q V R)
4 V 7
T
T
T
T
F
T
T
T
T
T
F
T
F
F
F
T
T
F
T
F
T
T
T
T
T
F
F
F
T
T
T
T
F
T
T
F
T
F
F
F
F
T
F
F
F
F
F
F
F
F
T
F
T
T
F
F
F
F
F
F
T
T
F
F
Possible switch setting
P
Q
R
Closed
Closed
Closed
Closed
Closed
Open
Closed
Open
Closed
Closed
Open
open
Example
Without using a truth table draw a sample network for the statement
(P Q) V [ P (~ Q V R)]
Solution
(P Q) V [ P (~ Q V R) ] = P (Q V (~ Q V R) ….. distributive
= P (Q V ~ Q) V R ……. associative
= P (t V R) ….. Complement
= P t ….. Identity
= P ….. Identity

The statement simplifies to p
The corresponding network is a follows
2Nsmjhs84Hipqosmuuogri7L9N5Kb6D89N Iwxazvik Lbhyb 86Dv9L9Rypjhptm5Oipstzr1Xa7Iw2Vw8Vs4Ni0Bcrqevt8Pasqgunrj2J7L7Wuo3J0Ldwlxjvymp T7Dtmso
For a statement which on simplifying ends upon F network drawn is as follows
Piczce2Sdpzjwi7Jr3Sobmo1H5Jdjbozd1Cstkyr40Htz
For a statement which upon simplifying yields to T, network is drawn as follows
7Kiay4Retlzk6Eeat75H11Kdzuvuzsbhwt Llm9Kyxhtxtgma9816Zsueq3Rxwjprwexbzqihgfjd6XhhtdqhykavcbpQUESTION
1. For each of the network shown below. Find a compound statement that represents it
Me 8T Qom Ofo15E0Ppg1R92Ul0Wkfdr5F2Hsuuav6Bgglczhawjryjla2Wdr03J8Dzbp1L0Bn5T927 8E1T56O78 7Xcjlpupiiqivgdsi3Ocp Mdr0Wjgrouih1Pcndqpz4Em




Swjq Rulwcn6 Olrr7Rnzzulgxfoo0Sr6Znifgyv3Ga4K3Cokjzsfojet1Cjw9Ytmcewevbuxlzktyibkrt9L8Yc5Uclbfgwlwu5Zyzlerwbiffkwknvth Kcunbv6Ddx 9T7Vq
2. (a) Draw network for the corresponding statement
i) (P ~ Q) (Q P)
ii) (P ~ Q) (Q ~ R)
iii) P → Q ≡ ~ P Q
iv) (P → Q) (p v Q) ≡ (~ P V Q) (P V Q)
(b) Simplify the statement in 2 (iv) using the laws of algebra of propositions and draw a simple network
MORE QUESTIONS
i) Write down compound statement for the following networks



Oq0G Pmm4Siltxf3Zjse 7Cmrdqiixi1Lwglf5Pueebnbisy 6Opcpsiwepif D3Zchz2 C6M2Rvzepxk1Boep99Gua7Xtpelbpepcja7G5Jyz Mtg6Stis


9Qu0Kriplmirrhbruagifebm7Qughzbx9J1Wbpebwf58 8Kcaxynhvqis37Wbkjw0Ewvjq3Kh6G5 L9Mssbz
Wwzaffvmcfqj1Onlzgiexg0Btos2Npqmkjjiilbse5I1Mohcmh4Kwvapoz Qsxral9J8B5E7Ptij86Ngtz53Z9 J2Gxvtkcjre3Knzo8Pqm4Pvbb5977Aagrjp8Tf8Jp B8Cpa

2. For each of these sentences draw a simple network
a) P(~ Q → ~p)
b) ~ (P Q) →R
c) P ~ P
3. Given a truth table
P
Q
R
T
T
T
F
T
T
F
T
T
F
T
T
T
F
F
F
F
T
T
T
F
T
F
F
F
F
T
F
F
F
F
T
a) Construct a statement having this truth table

b) Draw the electrical network

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TOPIC 3: LOGIC (I) ~ ADV MATHEMATICS FORM 5

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