Chapter - 1, Sentential Logic

Section - 1.4 - Operations on Sets


Summary

  • We can see a correspondence between set operations and statements. is the truth set of and is the truth set of then
    • Truth set of is .
    • Truth set of is .
    • Truth set of is .
  • Using logical forms of the statement, identities corresponding to set can be derived. For eg: is equivalent to can be proved as:


    Using distributive law,

    Using definition of

    Using definition of

    The above identity is similar to distributive law.

  • . This can also be proved similarly as above.
  • Set identities can also be proved using venn diagrams. One more way to prove them is by using truth tables also.
  • Sets and are disjoint .

Solutions

Soln1

(a) .

(b) .

(c) .

No sets from above are disjoint. Set (a) and (b) both are subsets of (c).


Soln2

(a) United States, Germany, China, Australia, France, India, Brazil .

(b) .

(c) .

We can see from above (c) is subset of (a). Also (b), (a) and (b), (c) are disjoint sets.


Soln3

It can be easily seen that both covers the same region.


Soln4

(a)

This can be verified easily from above diagram that .

(b)

This can be verified easily from above diagram that .


Soln5

Proof for first part(soln4a):

is equivalent to:

Using Demorgan’s Law,

Using Distribution Law,

As is a contradiction,

Using definition of set minus,

Hence proved.

Proof for second part(soln4b):



Using Distributive Law,

Using definition of

Using definition of

Hence Proved.


Soln6

(a)

This can be verified from above diagram that .

(b)

This can be verified from above diagram that .


Soln7

(a)
LHS:


Using Distributive Law,

Using commutative law,

Using definition of

Using definition of

RHS:


Using distributive law,

Using Absorption law,

Using Commutative law,

Using distributive law,

Using associative law,

Using Idempotent law,

Using definition of

Using definition of

From above we can see that after simplification LHS and RHS are same.

(b)

LHS:


Using Distributive Law,

Using Double Negation law,

Using Demorgans Law,

Using defn. of

Using defn. of

Using defn. of

= RHS. Hence Proved.


Soln8

(a)


(b)


Using Demorgans Law,

Using Demorgans Law,


(c)

.

(d)





(e)



As can be seen above a, d and e are equivalent. Also b, and c are equivalent.


Soln9

  1. , then .

  2. , then .


Soln10

(a) In the venn diagram given, there is no region corresponding to .

(b) First draw venn diagram for A, B and C using circles. Then draw a curve for D such that there is a region for every possible set that have possibility to contain some elements.

Soln11

(a)

It can be seen from above diagram that .

(b)

Let .

.

.


Soln12

It can be verified that both , and are equivalent as they corresponds to same region.


Soln13

For simplicity, I shall be using as logical statements corresponding to .

(a)

RHS:

















= LHS.

(b)

RHS:













= LHS.

(c)

RHS:












= LHS.

Skipping 14 and 15 as they are similar to 13.