Set Theory and Propositional Logic

This text explains the concepts of set theory and propositional logic. It covers topics like reflexive, symmetric, and transitive relations, and the use of quantifiers and connectives in propositional logic. Solved examples are provided for better understanding.

Q.a)
i. and
Venn Diagram
B ∪ C = { 5 } ∪ { 9 } ∪ { 3 } ∪ { 2,6,7 }
= {2,3,5,6,7,9 }
( A ∪ B)∩C = {5} ∩ {2,6,7} = ∅

ii.
1
2
3
4
.
.
.
n
For set of {1,2,3,4,5}
Let C = {1, 2, 3, 4, ..., n}
R = {(1,1), (2,1), (2,2), (3,1), (3,2), (3,3), (4,1), (4,2), (4,3), (4,4), ..., (n,1),
(n,2), ..., (n,n)}
Refexive:
Relation R is refexive.
Reason: Because (i,i) ∈ R where i={1,2,3, ..., n}.
Example: (1,1) ∈ R , (2,2) ∈ R, (3,3) ∈ R
Symmetric:
R is not symmetric.

Reason: Because if ∀(i,j) ∈ R there is ∃( j , i)∉ R
Example: (1,2) ∈ R but ( 2,1 ) ∉ R.
Transitive:
R is transitive.
Reason: Transitive property: if a=b and b=c then a=c.Here the
property holds true: if (i,j) ∈ R and (j,k) ∈ R then (i,k)∈ R. Since, j <= i
and k<= j then i <= k. Therfore relation (i,k) exists in the set R.
Example: (4,2) ∈ R and (2,1) ∈ R then (4,1) ∈ R
Q.b)
B(p) : printer ‘p’ is busy
Q(j) : print job ‘j’ is queued
i. If every printer is busy, then there is a job in queue:
( ∀ p B ( p ) ) →(∃ jQ( j))
The Proposition:
1. printer is busy
2. job is in queue
The quantifiers:
1. ∀ : For all
2. ∃ : there exists
The connectives:
1. : negation
2. → : if ... then
ii. Converse:
In words: if there is a queued job, then every printer is busy.
In symbol: (∃ jQ( j))→ ( ∀ p B ( p ) )
Contrapositive:

In words: ifc there is no queued job, then every printer is not busy.
In symbol: (∃ j Q( j)) → ( ∀ p B ( p ) )
Simplified Contrapositive:
In words: If for all jobs are not queued, then there exists a printer
which is not busy.
In symbol: ( ∀ j Q( j)) → ( ∃ p B ( p ) )
Negation:
In words: If every printer is busy, then there does not exist a job
which is queued.
In symbols: ( ∀ p B ( p ) ) → (∃ jQ( j))
Simplified Negation:
In words: If every printer is busy, then for every job no one is queued.
In symbol: ( ∀ p B ( p ) ) →( ∀ j Q ( j))