LOGIC

1. LOGIC
Logic concerns with the study of analysis of methods of reasoning which lead to certain
conclusion or statement.
Example
If it rains today we shall play football. It rained we did not play football
Simple andcompound sentence
Consider the followingsentence
i) The medians of a triangle meet at a point
ii) The diagonals of any quadrilateral are parallel
i) and ii) are simple sentence
iii) The median of a triangle meet at a point and the diagonal of a quadrilateral are parallel
iii) Is a compoundsentence
Other connectingwords are: or, but, while
TRUTH VALUE OF A SENTENCE
2 x 3 = 5 has a truthvalue " false"
2 x 2 = 4 has a truthvalue " True"
The number 23 is prime. "True"
Propositions
A proposition is any statement which is free from ambiguity and having a property, it is
either true or false but not both nor neither.
Consider the followingsentence;
i) Birds have no wings (p)
ii) The sun rises from the west. (p)

2. iii) 8 = 6 + 2 (p)
iv) The grass is green(p)
v) I am feelinghungry (not proposition)
TRUTH TABLE
A truth table is a matrix whose entries are truthvalues
E.g. F T Or T F
A completetruthfor conjunction
Note: A conjunctionis a compound propositionconnectedbythe word "and"
E.g. the sun rises from the west and 8 = 6 + 2
The above propositionhas a truth value false
The proposition having and/ but, if both of the sentence are true, then only truth value will be
true.
The word “but” carries the same meaning as the word “and”
QUESTIONS
Find the components or simple sentenceof the followingconjunctions
a) 3< 5 and three are infinitelymany prime numbers.
i) 3< 5
ii) There are infinitelymany prime numbers
b) 4 is divisible by 2 and 4 is a prime number
i) 4 is divisible by 2
T T F F
T F T F
F F F F
T T F F

3. ii) 4 is a prime number
c) 2 < 3 and 5 < 3
i) 2< 3
ii) 5< 3
d) The sun rises from the west and is irrational
i) The sun rises from the west
ii) is irrational
e) 2 is an odd number and it is false that 5 is even
A completetruthtable for conjunction
Let P and Q be any general proposition
Requiredto find the truth table for P and Q
Now P and Q is writtenas P ∧ Q
P ∧ Q has truthvalue onlywhen bothP and Q are true
Truth table for P ∧ Q
Negation
A negation is a sentence which has an opposite truthvalue to the given one
P Q P Q
T T T
T F F
F T F
F F F

4. - One way of forminga negation is to put the word ‘’ not’’ with a verb
Example: 6 is divisible by 3
6 is not divisible by 3
It is not true that 6 is divisible by 3
It is false that 6 is divisible by 3
Given a statement P, its negation is denoted P
The complement of ~ P is P
Truth Table for negation
P ~P
T F
F T
Disjunction
Another word used to combine sentence is the word " or "
Consider the sentence
i) 43 < 3 ii) 5 > 3
- combiningthem with the word " or " i.e. 43 < 3 or 5 > 3
- The connective word " or " is calleda disjunctionand is symbolizedby " V "
The truth value for disjunctionis onlyfalse when both the components are false
If P and Q are statement,thenP or Q is symbolizedas P V Q
P V Q has a truthvalue false in one case when both P and Q are false
Truth Table for disjunction

5. P Q P V Q
T T T
T F T
F T T
F F F
Implications
These are statements of the form "if……..then……"
Example. If a quadrilateral is a parallelogram thenthe pair of opposite sides are parallel
The phrase "a quadrilateral is a parallelogram" calledhypothesis or antecedent
The phrase "the pair of opposite sides are parallel"is calleda conclusionor constituent
If P hypothesis
Q conclusion
Then the statement if P then Q its implicationin short we write P Q
Consider the statement
If 43 < 3 then 5 > 3 T
If 43 < 3 then 5 < 3 T If hypothesis is T and conclusion is F 43 > 3 then 5 <
3 F then the implicationis T
→Note: The compound statement P → Q is false only in one case P is true and Q is false.
Truth Table for P Q
P Q P → Q
T T T

6. T F F
F T T
F F T
Propositions whichcarrythe same meaning as if P then Q
i) If P, Q
ii) Q if P
iii) Q provided that P
iv) P only if Q
v) P is a sufficient conditionfor Q
vi) Q is a necessaryconditionfor P
EXERCISE
1. Determine the truthvalues of the following
a) If 2 < 3 then 2 + 3 = 5 T
b) If 3 < 2 then 3 + 2 = 5 T
c) If 2 + 3 = 5 then 3 < 2 F
d) If 2 + 1 = 2 then 1 = 0 T
2. Find the components of the followingcompound
i) If 3 < 5 then 10 + m = 9
a) 3 < 5
b) 10 + m = 9
ii) a + b = c + d only if p + q = r2

7. a) a + b = c + d
b) p + q = r2
iii) If Galileo was born before Descartes thenNewtonwas bornbefore Shakespeare
a) Galileo was bornbefore Descartes
b) Newton was born before Shakespeare
3. Write a truthtable for
i) (P ∧ Q) V (P V Q) ii) (P → Q) ∧ P iii) ((P → Q) → Q)
Solutions
i) (P ∧ Q) V (P V Q)
ii) (P → Q) ∧ P
P Q P → Q (P → Q)∧ P
T T T T
T F F F
F T T F
F F F F
P Q P ∧ Q P V Q (P ∧ Q) V (P V Q)
T T T T T
T F F T T
F T F T T
F F F F F

8. iii) ((P → Q) → Q)
P Q P → Q (P → Q) → Q
T T T T
T F F T
F T T T
F F T F
BI CONDITIONAL STATEMENT
Consider the truth table for (P → Q) ∧ (Q → P)
(A) (B)
P Q P → Q Q → P A ∧ B
T T T T T
T F F T F
F T T F F
F F T T T
The statement (P → Q) ∧ (Q → P) is known as bi-conditional statement and is abbreviated as
P Q
Truth table for P Q
P Q P Q
T T T
T F F
F T F

9. F F T
Note: P Q is read P if and only if Q
P Q is true when both P and Q are true or when P and Q are false
Example. The truth value of 43 > 3 if and only if 5< 3 (F)
43 < 3 if and onlyif 3 < 5 (F)
43< 3 if and only if 5 < 3 (T)
43> 3 if and only if 5 > 3 (T)
CONVERSE, CONTRA POSITIVE, INVERSE
Given a proposition: if a quadrilateral is a parallelogram then its opposite sides are parallel, P
→ Q
Converse: If the opposite sides are parallel, then the quadrilateral is a parallelogram i.e. Q
→ P.
Contra positive: If the positive sides are not parallel, then the quadrilateral is not a
parallelogram. i.e. ~ Q → ~ P
Inverse: if a quadrilateral is not a parallelogram, then the opposite sides are not parallel i.e. ~
P → ~ Q
Truth table for implication, converse, contrapositive, inverse
P Q P → Q Q → P ~ P ~ Q ~ Q → ~P ~P → ~ Q
T T T T F F T T
T F F T F T F T
F T T F T F T F
F F T T T T T T
Column 3 has exactlytruth value as column7
i. e P → Q ~Q → ~ P
Q → P ~ P → ~Q

10. EQUIVALENT STATEMENTS
Two propositions are logicallyequivalent if they have exactlythe same truth values
E.g. P V Q and Q V P are logicallyequivalent
Solution: Draw truth for P V Q and Q V P
1 2 3 4
P Q P V Q Q V P
T T T T
T F T T
F T T T
F F F F
Since column3 has exactly the same truthvalues as column4 then
P V Q Q V P
Questions
Show whether or not the following propositions are logically equivalent
i)P → Q, ~ P V Q
P Q P → Q ~ P ~ P V Q
T T T F T
T F F F F
F T T T T
F F T T T
Since column 3 and 5 have exactly the same truth value therefore
P → Q ~ P V Q
ii) P → (P V Q); P → Q
P Q P V Q 1 → 3 p → Q

11. Since column 4 does not have exactly same truth value as column 5 then p → (P V Q) P
→ Q
iii) P → Q: ~ P → Q
P Q P → Q ~ P ~ P → Q
T T T F T
T F F F T
F T T T T
F F T T F
Since column3 does not have exactlysame truthvalues as column5 therefore
P → Q ~ P → Q
iv) P → Q; Q → P
P Q P → Q Q → P
T T T T
T F F T
F T T F
F F T T
T T T T T
T F T T F
F T T T T
F F F T T

12. Since column 3 does not have exactly same truth values as column 4 therefore P → Q Q
→ P
v) ~ (P → Q);PV ~ Q
(5) (6)
P Q ~ Q P → Q ~ (P → Q) P V ~Q
T T F T F T
T F T F T T
F T F T F F
F F T T F T
Since column 5 does not have exactly same truth value as column 6 therefore ~ (P → Q) P
V ~Q
vi) ~ (P V Q); ~P ∧ ~Q
P Q ~ P ~ Q P V Q ~ (P V Q) ~P ∧ ~Q
T T F F T F F
T F F T T F F
F T T F T F F
F F T T F T T
Since column6 has exact same truth values as column7 therefore ~ (P V Q) →( ~P ∧ ~Q)
COMPOUND STATEMENTS
Compound statements with three components P, Q, R.
Consider the followingcompoundstatement,
Triangles have all three sides and either the area of a circular region of radius r is or it is
false that the diagonals of a parallelogram do not meet.
Solution
(To symbolize the above statement)

13. Let P triangles have three sides
Let Q circular regionof radius r is
Let R diagonals of parallelogram do not meet
P ∧ (Q V ~R)
To find the truth values of the above statement
P Q R ~ R Q V ~R P ∧ (Q V ~ R)
T T F T T T
 The statement has a truth value true
TAUTOLOGY
A tautology is a proposition which is always true under all possible truth conditions of its
component parts
Example
Show that whether or not ~ (P ∧ Q) V (~P → ~Q) is a tautology
(6) (7)
P Q ~ P ~ Q P ∧ Q ~ (P ∧ Q) ~ P → ~Q 6 V 7
T T F F T F T T
T F F T F T T T
F T T F F T F T
F F T T F T T T
Since column8 has all the truthvalues True (T) therefore it is TAUTOLOGY
Since column8 has truth value true throughout then, ~ (P ∧ Q) V (~ P → ~Q) is a tautology

14. Questions
1. Show whether the given compoundstatements are tautologyor not
i) (P ∧ Q) → P
P Q P ∧ Q (P ∧ Q) → P
T T T T
T F F T
F T F T
F F F T
Since column4 has truth value true throughout then (P ∧ Q) → P is a tautology.
ii) P → (P ∧ Q)
Since column 4 does not have truth value true throughout then P → (P ∧ Q) is not a
tautology.
iii) P → ~P
P Q (P ∧ Q) P → (P ∧ Q)
T T T T
T F F F
F T F T
F F F T
P ~ P P → ~P
T F F
F T T

15. Since column3 does not have the truth value true throughout then
P → ~ P is not a tautology.
iv) (P → Q) → (~ P → Q)
P Q P → Q ~ P ~P → Q (P → Q) → (~ P → Q)
T T T F T T
T F F F T T
F T T T T T
F F T T F F
Since column 6 does not have the truth value true throughout then (P → Q) →(~ P → Q) is
not a tautology
v) (P → Q) V (Q → P)
P Q P → Q Q → P 3 V 4
T T T T T
T F F T T
F T T F T
F F T T T
Since column 5 has all truth values true throughout then (P → Q) V (Q → P) is a
tautology.
2. Express the followingin symbolic form and then find its truthvalue
i) 2 is a prime, and either 4 is even or it’s not true that 5 is even
Solution
Let P 2 is a prime
Let Q 4 is even

16. Let R 5 is even
P ∧ (Q V ~R)
P Q R ~ R Q V ~ R P ∧ (Q V ~ R)
T T T F T T
P ∧ (Q V ~R) has a truthvalue true.
ii) 7 is odd, or either Londonis in France and it is false that Paris is not in Denmark
Let P 7 is odd
Q London is in France
R Paris is not in Denmark
P V (Q ∧ ~R)
P Q R ~ R Q ∧ ~R P V (Q ∧ ~ R)
T F T F F T
P V (Q ∧ ~R) has a truth value True
3. Find the truth values of P ∧ (Q V ~R) if
i) P, Q, R all has truth value T
ii) If P, Q, R all have truth value of F
iii) If P is true, Q is false and R is false
P Q R ~ R Q V ~ R P Λ (Q V ~
R)
T T T F T T

17. F F F T T F
T F F T T T
A complete truthtable for general cases
1. Only one compound P
Two rows
2. Two components P and Q
Four rows
3. Three components P, Q, R
Eight Rows
P
T
F
P Q
T T
T F
F T
F F
P Q R
T T T
T T F

18. 4. Four components P, Q, R, S
Sixteenrows
P Q R S
T T T T
T T T F
T T F T
T T F F
T F T T
T F T F
T F T
T F F
F T T
F T F
F F T
F F F

19. T F F T
T F F F
F T T T
F T T F
F T F T
F T F F
F F T T
F F T F
F F F T
F F F
Example constructs atruthtable for the compound statement
((P → Q) Λ R) Q
P Q R P → Q (P → Q) Λ R 5 2
T T T T T T
T T F T F F
T F T F F T
T F F F F T
F T T T T T
F T F T F F

20. F F T T T F
F F F T F T
LAWS OF ALGEBRA OF PROPOSITIONS
1. Idempotent laws
a) P V P P
b) P Λ P P
2. Commutative
a) P V Q Q V P
b) P Q Q Λ P
3. Associative laws
a) (P V Q) V R P V (Q V R)
b) (P Λ Q) Λ R P Λ (Q Λ R)
4. Distributive laws
a) P V (Q Λ R) (P V Q) Λ (P V R)
b) P Λ (Q V R) (P Λ Q) V (P Λ R)
5. Identitylaws
a) P V f P
b) P Λ t P
c) P V t t

21. d) P Λ f f
6. Complementarylaws
a) P V ~ P t
b) P Λ ~ P f
c) ~ ~P P
d) ~ T F or t~ f
e) ~ F T or f~ t
7. De-Morgan’s law
a) ~ (P V Q) ~ P Λ ~ Q
b) ~ (P Λ Q) ~ P V ~ Q
Examples
Using the laws of algebra of propositionsimplify (P V Q) ∧ ~ P
Solution
(P V Q) Λ ~ P (~ P Λ P) V (~ P Λ Q) ……distributive law
f V (~ P Λ Q) ………compliment law
(~ P Λ Q) ………..identity
Questions
1. Simplifythe followingpropositions usingthe laws of algebra of propositions
i) ~ (P V Q) V (~P Λ Q)
ii) (P Λ Q) V [~ R Λ (Q Λ P)]
2. Show using the laws of algebra of propositions (P Λ Q) V [P Λ (~Q V R)] P

22. 3. Construct atruth table for [(p → ~q) ∧ (r → p) ∧ r] → ~p
SENTENCE HAVING A GIVEN TRUTH TABLE
Example.1 Find a sentence which has the followingtruthtable
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 conjunctionof P and Q
3. Requiredsentence is the disjunctions of the above basic conjunction
Requiredsentence (P ∧ Q) V (P ∧ ~ Q) V (~ P ∧ Q)
Example. 2
P Q - Basic conjunction
T T T P ∧ Q
T F T P ∧ ~Q
F T T ~ P ∧ Q
F F F

23. 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
Requiredsentence is (P ∧ Q ∧ R) V (P ∧ ~ Q ∧ ~ R)
Example. 3
Find a sentence having the followingtruth table and simplifyit.
P Q R - Basic conjunction
T T T T P ∧ Q∧ R
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

24. 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 requiredsentence 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

25. QUESTIONS
1. for each of the following truth tables (a), (b) and (c) construct a compound sentence
having that truthtable.
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

26. F T F F F F
F F T F F F
F F F F F T ~ P ∧ ~ Q ∧ ~ R
→The requiredsentence 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 requiredsentence for (c)is (P ∧ Q ∧ ~R) V (P ∧ ~Q ∧ R) V (~P ∧ ~Q ∧ ~R)
2. i) construct atruth table for ~ (P → Q)
ii) Write a compoundsentence having that truthtable (involving ~, ∧ , v)
3. Repeat for the followingsentence
i) ~ P → ~Q ii) ~ p 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 compoundsentence 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 q and ~ p v q are logicallyequivalent

27. 7. If Apq p ∧ q and Np ~ p write the followingwithout ~ and A
i) ~ (p ∧ q)
ii)~ (p ∧ ~q)
iii) ~ (~ p ∧ q)
iv) ~ (p ∧ ~ q)
QUESTIONS
1. Rewrite the followingwithout using the conditional
i) If it is cold, he wears a hat
ii) If productivity increases, thenwages rise
2. Determine the truthvalue of the following
i) 2 + 2 = 4 if and onlyif 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 truthtable
i) ~ (p q) ≡ p ~q
ii) ~ (p q) ≡ ~ p q
4. Prove the conditional distributes over conjunctioni.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.

28. ii) A necessaryconditionfor it to be coldis that it rains.
iii) A sufficient conditionfor it to be coldis 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 contrapositive of
"If the diagonals of the rhombus are perpendicular then it is a square"
LOGICAL IMPLICATIONS
A propositionP is saidto be logicallyimply a propositionQ if p → Q is a tautology
Example
Show that p logicallyimplies pv q
Solution;Construct a truth table for p → (p v q)
Since column4 is a tautologythen p logicallyimplies p v q
ARGUMENTS
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

29. 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
denotedby p1, p2….pn Q
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 determinedas follows
→An argument P1, P2, P3… Pn 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 followingargument is valid or not P, P → Q 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

30. 1. Since in row 1 the conclusion is true and all the premises are true then the argument is
valid
2. Since column5 is a tautologythen the argument is valid
QUESTION
Use the truthtable to show whether the given argument is valid or not
P → Q, Q → R 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 symbolizedas 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, ~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

31. 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 column8 is not a tautologythe given argument is not valid
QUESTIONS
1. Translate the followingarguments 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,
thereforeLondonis in Denmark
ii) If I work I cannot study. Either I work or I pass mathematics. I passed mathematics
thereforeIstudied.
iii) If I buy books, I lose money. I bought books, thereforeIlost money
2. Determine the validity of
i) p → q, ~q ~p
ii)~p → q, p ~q
iii) [p → ~ q], r → q, r ~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

32. When the switchp is closedthe current flows betweenT1 and T2
The above network simplifies to the following network
Relationship between statement in logic and network
A SERIES AND PARALLEL CONNECTION OF SWITCHES
A seriesconnectionofswitches
The followingswitches are connectedinseries

33. The current flow between T1 and T2 when both switches are closed current flows
when p ∧ Q is true
A parallel connectionofswitches
The current will flowwhen either one of the switches is closed.
Currents flowwhen P V Q is true
Example
Consider the electrical networkbelow
i) Construct acompound statement presentingthe networkabove
ii) Find possible switch setting that will allow the current to flow between T1 and
T2

34. Solution
Note i) current flows betweenT1 and T2 when switch p is closedi.e. p is true OR
ii) The current flows between T1 and T2 when switch switches q and r are
closedi.e. Q ∧ R is true.
The requiredcompoundstatement is p v (Q ∧ R)
iii) To find possible switchsetting, 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 switchsetting
P Q r
Closed Closed Closed
Closed Closed Closed
Closed Closed Closed
Closed Closed Closed

35. Closed Closed Closed
Questions
1. Construct compoundstatement that correspondto the networks
Solution
The current will flowwhen all three switches p, q, and r are closedi.e. p ∧ q ∧ r
The requiredcompoundstatement is P ∧ Q ∧ R
The requiredcomponent statement is (P ∨ q)
The requiredcompoundstatement is (p ∧ q) V (r∧ s)

36. The requiredcompoundstatement is P V Q V R
The requiredcompoundstatement is p ∧ (q V (r ∧ s))
The requiredcompoundstatement 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

37. Possible switchsettings
From statements to network
Example
Draw a network for the statement (pv Q) ∧ (R ∧ S)
Solutions
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
P Q R
Closed Closed Closed
Closed Open closed

38. Correspondingnetworkis shown below
Questions
Draw network for the followingstatements
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 switchis closed, the other one closes also
ii) When one switch is closed, the other one opens
Refers to the diagram

39. The compound relatingto flow of electrical current is given
(P ∧ Q) V [P ∧ (~ Q V R)]
To find possible switchsettingthat will allow the current to flowbetween 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

40. 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 switchsetting
P Q R
Closed Closed Closed
Closed Closed Open
Closed Open Closed
Closed Open open
Example
Without using a truthtable 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

41. = P ∧ t ….. Identity
= P ….. Identity
The statement simplifies to p
The correspondingnetworkis a follows
For a statement which on simplifyingends upon F network drawn is as follows
For a statement which upon simplifyingyields to T, network is drawn as follows
QUESTION
1. For each of the network shown below. Find a compound statement that represents it

42. 2. (a) Draw network for the correspondingstatement
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 compoundstatement for the followingnetworks

43. 2. For each of these sentences drawa simple network
a) P∧ (~ Q → ~p)
b) ~ (P ∨ Q) →R

44. c) P ∧ ~ P
3. Given a truth table
a) Construct a statement having this truth table
b) Draw the electrical network
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