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MATHEMATICS FORM FIVE TOPIC 3: LOGIC (II)

 

 SENTENCE HAVING A GIVEN TRUTH TABLE

Example.1 Find a sentence which has the following truth table

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

T

T

T

T

F

T

F

T

T

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

Basic conjunction

T

T

T

P Q

T

F

T

P ~Q

F

T

T

~ P Q

F

F

F


Solution

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

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

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


SOLUTION

P

Q

T

T

T

T

F

F

F

T

T

F

F

T

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

Basic conjunction

T

T

T

P Q

T

F

F


F

T

T

~ P Q

F

F

T

~ P ~Q

Solution

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

→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  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  q and ~ p v q are logically equivalent

7. If Apq  p q and Np ~ 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  q) ≡ p  ~q

ii) ~ (p  q) ≡ ~ p  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

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

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 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 determined as 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 following argument is valid or not P, P → Q Q

Solution:

Draw a truth table for [P P → Q] → 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

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  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, ~p

Testing the validity

[(p → q) (q r)r] → ~p

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

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  ~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

When the switch p is closed the current flows between T1 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 series connection of switches

The following switches are connected in series

The current flow between T1 and T2 when both switches are closed current flows when p Q is true

A parallel connection of switches

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

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

Possible switch setting

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

Questions

1. Construct compound statement that correspond to the networks

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

The required component statement is (P q)




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



The required compound statement is P V Q V R

The required compound statement is p (q V (r s))





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

Closed

Closed

Closed

Closed

Closed

Closed

Closed

Closed

Closed

Closed

Closed

Closed

Closed

Closed

Closed




Possible switch settings

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

From statements to network

Example

Draw a network for the statement (p v Q) (R S)

Solutions

Corresponding network is shown below

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

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

Closed

Closed

Closed

Closed

Open

closed

Possible switch setting

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

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


For a statement which on simplifying ends upon F network drawn is as follows

For a statement which upon simplifying yields to T, network is drawn as follows
QUESTION

1. For each of the network shown below. Find a compound statement that represents it






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









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

Closed

Closed

Closed

Closed

Closed

Open

Closed

Open

Closed

Closed

Open

open

a) Construct a statement having this truth table

b) Draw the electrical network

 

Mathematics (from Ancient Greek μάθημα; máthēma: ‘knowledge, study, learning’) is an area of knowledge that includes such topics as numbers (arithmetic, number theory), formulas and related structures (algebra), shapes and the spaces in which they are contained (geometry), and quantities and their changes (calculus and analysis).

Most mathematical activity involves the use of pure reason to discover or prove the properties of abstract objects, which consist of either abstractions from nature or—in modern mathematics—entities that are stipulated with certain properties, called axioms. A mathematical proof consists of a succession of applications of some deductive rules to already known results, including previously proved theorems, axioms and (in case of abstraction from nature) some basic properties that are considered as true starting points of the theory under consideration.

Mathematics is used in science for modeling phenomena, which then allows predictions to be made from experimental laws. The independence of mathematical truth from any experimentation implies that the accuracy of such predictions depends only on the adequacy of the model. Inaccurate predictions, rather than being caused by incorrect mathematics, imply the need to change the mathematical model used. For example, the perihelion precession of Mercury could only be explained after the emergence of Einstein’s general relativity, which replaced Newton’s law of gravitation as a better mathematical model.

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Q

R

T

T

T

F

T

T

F

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T

T

T

F

F

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