ADVANCED MATHEMATICS FORM 5 – LOGIC

            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.   Identity laws

            a) P V f  P

            b) P Λ t P

            c) P V t t

            d) P Λ f f

6.   Complementary laws

            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 proposition simplify (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.  Simplify the following propositions using the 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

3.  Construct a truth table for   [(p → ~q) ∧ (r → p) ∧ r] → ~p

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

Required sentence (P ∧ Q) V (P ∧ ~ Q) V (~ P ∧ Q)

Example. 2

Find a sentence having the truth table below

        Solution

 

Required sentence is (P ∧ Q ∧ R) V (P ∧ ~ Q ∧ ~ R)

Example. 3

            Find a sentence having the following truth table and simplify it.

SOLUTION

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.

Solution

→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”