**Chapter 1 Mathematical Logic Ex 1.5**

## Chapter 1 Mathematical Logic Ex 1.5

**Question 1.Express the following circuits in the symbolic form of logic and writ the input-output table.(i)**

Solution:

Solution:

Let p : the switch S

_{1}is closed

q : the switch S

_{2}is closed

r : the switch S

_{3}is closed

~p : the switch S

_{1}â€˜ is closed or the switch S

_{1}is open

~q : the switch S

_{2}â€˜ is closed or the switch S

_{2}is open

~r : the switch S

_{3}â€˜ is closed or the switch S

_{3}is open

l : the lamp L is on

(i) The symbolic form of the given circuit is : p âˆ¨ (q âˆ§ r) = l

l is generally dropped and it can be expressed as : p âˆ¨ (q âˆ§ r).

**(ii)Solution:**
The symbolic form of the given circuit is : (~ p âˆ§ q) âˆ¨ (p âˆ§ ~ q).

**(iii)Solution:**

The symbolic form of the given circuit is : [p âˆ§ (~q âˆ¨ r)] âˆ¨ (~q âˆ§ ~ r).

**(iv)Solution:**

The symbolic form of the given circuit is : (p âˆ¨ q) âˆ§ q âˆ§ (r âˆ¨ ~p).

**(v)Solution:**

The symbolic form of the given circuit is : [p âˆ¨ (~p âˆ§ ~q)] âˆ¨ (p âˆ§ q).

**(vi)Solution:**
The symbolic form of the given circuit is : (p âˆ¨ q) âˆ§ (q âˆ¨ r) âˆ§ (r âˆ¨ p)

**Question 2.Construct the switching circuit of the following :(i) (~pâˆ§ q) âˆ¨ (pâˆ§ ~r)Solution:**

Let p : the switch S

_{1}is closed

q : the switch S

_{2}is closed

r : the switch S

_{3}is closed

~p : the switch S

_{1}â€˜ is closed or the switch S

_{1}is open

~ q : the switch S

_{2}â€˜ is closed or the switch S

_{2}is open

~ r : the switch S

_{3}â€˜ is closed or the switch S

_{3}is open.

Then the switching circuits corresponding to the given statement patterns are :

**(ii) (pâˆ§ q) âˆ¨ [~p âˆ§ (~q âˆ¨ p âˆ¨ r)]Solution:**

**(iii) [(p âˆ§ r) âˆ¨ (~q âˆ§ ~r)] âˆ§ (~p âˆ§ ~r)Solution:**

**(iv) (p âˆ§ ~q âˆ§ r) âˆ¨ [p âˆ§ (~q âˆ¨ ~r)]Solution:**

**(v) p âˆ¨ (~p ) âˆ¨ (~q) âˆ¨ (p âˆ§ q)Solution:**

**(vi) (p âˆ§ q) âˆ¨ (~p) âˆ¨ (p âˆ§ ~q)Solution:**

**Question 3.Give an alternative equivalent simple circuits for the following circuits :(i)Solution:**

(i) Let p : the switch S

_{1}is closed

q : the switch S

_{2}is closed

~ p : the switch S

_{1}â€˜ is closed or the switch Si is open Then the symbolic form of the given circuit is :

p âˆ§ (~p âˆ¨ q).

Using the laws of logic, we have,

p âˆ§ (~p âˆ¨ q)

= (p âˆ§ ~ p) âˆ¨ (p âˆ§ q) â€¦(By Distributive Law)

= F âˆ¨ (p âˆ§ q) â€¦ (By Complement Law)

= p âˆ§ qâ€¦ (By Identity Law)

Hence, the alternative equivalent simple circuit is :

**(ii)**

Let p : the switch S

_{1}is closed

q : the switch S

_{2}is closed

r : the switch S

_{3}is closed

~q : the switch S

_{2}â€˜ is closed or the switch S

_{2}is open

~r : the switch S

_{3}â€˜ is closed or the switch S

_{3}is open.

Then the symbolic form of the given circuit is :

[p âˆ§ (q âˆ¨ r)] âˆ¨ (~r âˆ§ ~q âˆ§ p).

Using the laws of logic, we have

[p âˆ§ (q âˆ¨ r)] âˆ¨ (~r âˆ§ ~q âˆ§ p)

â‰¡ [p âˆ§ (q âˆ¨ r)] âˆ¨ [ ~(r âˆ¨ q) âˆ§ p] â€¦. (By De Morganâ€™s Law)

â‰¡ [p âˆ§ (q âˆ¨ r)] âˆ¨ [p âˆ§ ~(q âˆ¨ r)] â€¦ (By Commutative Law)

â‰¡ p âˆ§ [(q âˆ¨ r) âˆ¨ ~(q âˆ¨ r)) â€¦ (By Distributive Law)

â‰¡ p âˆ§ T â€¦ (By Complement Law)

â‰¡ p â€¦ (By Identity Law)

Hence, the alternative equivalent simple circuit is :

**Question 4.Write the symbolic form of the following switching circuits construct its switching table and interpret it.i)Solution:**

Let p : the switch S

_{1}is closed

q : the switch S

_{2}is closed

~p : the switch S

_{1}â€˜ is closed or the switch S

_{1}is open

~ q : the switch S

_{2}â€˜ is closed or the switch S

_{2}is open.

Then the symbolic form of the given circuit is :

(p âˆ¨ ~q) âˆ¨ (~p âˆ§ q)

Since the final column contains allâ€™ 1â€², the lamp will always glow irrespective of the status of switches.

**(ii)Solution:**
Let p : the switch S

_{1}is closed

q : the switch S

_{2}is closed

~p : the switch S

_{1}is closed or the switch S

_{1}is open.

~q : the switch S

_{2}â€˜ is closed or the switch S

_{2}is open.

Then the symbolic form of the given circuit is : p âˆ¨ (~p âˆ§ ~q) âˆ¨ (p âˆ§ q)

Since the final column contains â€˜0â€™ when p is 0 and q is â€˜1â€™, otherwise it contains â€˜1â€².

Hence, the lamp will not glow when S

_{1}is OFF and S

_{2}is ON, otherwise the lamp will glow.

**iii)Solution:**

Let p : the switch S

_{1}is closed

q : the switch S

_{2}is closed

r : the switch S

_{3}is closed

~q : the switch S

_{2}â€˜ is closed or the switch S

_{2}is open

~r: the switch S

_{3}â€˜ is closed or the switch S

_{3}is open.

Then the symbolic form of the given circuit is : [p âˆ¨ (~q) âˆ¨ r)] âˆ§ [p âˆ¨ (q âˆ§ r)]

From the switching table, the â€˜final columnâ€™ and the column of p are identical. Hence, the lamp will glow which S

_{1}is â€˜ONâ€™.

**Question 5.Obtain the simple logical expression of the following. Draw the corresponding switching circuit.(i) p âˆ¨ (q âˆ§ ~ q)Solution:**

Using the laws of logic, we have, p âˆ¨ (q âˆ§ ~q)

â‰¡ p âˆ¨ F â€¦ (By Complement Law)

â‰¡ p â€¦ (By Identity Law)

Hence, the simple logical expression of the given expression is p.

Let p : the switch S

_{1}is closed

Then the corresponding switching circuit is :

**(ii) (~p âˆ§ q) âˆ¨ (~p âˆ§ ~q) âˆ¨ (p âˆ§ ~q)]Solution:**
Using the laws of logic, we have,

(~p âˆ§ q) âˆ¨ (~p âˆ¨ ~q) âˆ¨ (p âˆ§ ~q)

â‰¡ [~p âˆ§ (q âˆ¨ ~q)] âˆ¨ (p âˆ§ ~ q)â€¦ (By Distributive Law)

â‰¡ (~p âˆ§ T) âˆ¨ (p âˆ§ ~q) â€¦ (By Complement Law)

â‰¡ ~p âˆ¨ (p âˆ§ ~q) â€¦ (By Identity Law)

â‰¡ (~p âˆ¨ p) âˆ§ (~p âˆ§~q) â€¦ (By Distributive Law)

â‰¡ T âˆ§ (~p âˆ§ ~q) â€¦ (By Complement Law)

â‰¡ ~p âˆ¨ ~q â€¦ (By Identity Law)

Hence, the simple logical expression of the given expression is ~ p âˆ¨ ~q.

Let p : the switch S

_{1}is closed

q : the switch S

_{2}is closed

~ p : the switch S

_{1}â€˜ is closed or the switch S

_{1}is open

~ q : the switch S

_{2}â€˜ is closed or the switch S

_{2}is open,

Then the corresponding switching circuit is :

**(iii) [p (âˆ¨ (~q) âˆ¨ ~r)] âˆ§ (p âˆ¨ (q âˆ§ r)Solution:**
Using the laws of logic, we have,

[p âˆ¨ (~ (q) âˆ¨ (~r)] âˆ§ [p âˆ¨ (q âˆ§ r)]

= [p âˆ¨ { ~(q âˆ§ r)}] âˆ§ [p âˆ¨ (q âˆ§ r)] â€¦ (By De Morganâ€™s Law)

= p âˆ¨ [~(q âˆ§ r) âˆ§ (q âˆ§ r) ] â€¦ (By Distributive Law)

= p âˆ¨ F â€¦ (By Complement Law)

= p â€¦ (By Identity Law)

Hence, the simple logical expression of the given expression is p.

Let p : the switch S

_{1}is closed

Then the corresponding switching circuit is :

**(iv) (p âˆ§ q âˆ§ ~p) âˆ¨ (~p âˆ§ q âˆ§ r) âˆ¨ (p âˆ§ ~q âˆ§ r) âˆ¨ (p âˆ§ q âˆ§ r)Question is Modified(p âˆ§ q âˆ§ ~p) âˆ¨ (~p âˆ§ q âˆ§ r)âˆ¨ (p âˆ§ q âˆ§ r)Solution:**
Using the laws of logic, we have,

(p âˆ§ q âˆ§ ~p) âˆ¨ (~p âˆ§ q âˆ§ r) âˆ¨ (p âˆ§ q âˆ§ r)

= (p âˆ§ ~p âˆ§ q) âˆ¨ (~p âˆ§ q âˆ§ r) âˆ¨ (p âˆ§ q âˆ§ r) â€¦ (By Commutative Law)

= (F âˆ§ q) âˆ¨ (~p âˆ§ q âˆ§ r) âˆ¨ (p âˆ§ q âˆ§ r) â€¦ (By Complement Law)

= F âˆ¨ (~p âˆ§ q âˆ§ r) âˆ¨ (p âˆ§ q âˆ§ r) â€¦ (By Identity Law)

= (~p âˆ§ q âˆ§ r) âˆ¨ (p âˆ§ q âˆ§ r) â€¦ (By Identity Law)

= (~ p âˆ¨ p) âˆ§ (q âˆ§ r) â€¦ (By Distributive Law)

= T âˆ§ (q âˆ§ r) â€¦ (By Complement Law)

= q âˆ§ r â€¦ (By Identity Law)

Hence, the simple logical expression of the given expression is q âˆ§ r.

Let q : the switch S

_{2}is closed

r : the switch S

_{3}is closed.

Then the corresponding switching circuit is :