ADVANCED MATHEMATICS FORM 5 – LOGIC

  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)

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….p_n called premises yields to another proposition Q called a conclusion such as argument is denoted by p₁, p₂….p_n
 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… P_n  Q is valid if Q is true whenever all the premises P₁, P₂, P₃… P_n are true

→Validity of an argument is also determined if and only if the proposition (P₁
∧ P₂
∧ P₃
∧ ….. P_n) → 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

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

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)

Possible switch setting

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 (r∧ s)

                       

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

 

Possible switch settings

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

            Possible switch setting

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