Download List all the partitions of A={1,2,3}

Discrete Structure
BY Prof.NILOY GANGULY
scribe prepared by Krishna Devaroy G.T ,07cs1012
25/08/08
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RELATION
List all the partitions of A={1,2,3}
1. P1 ={ {1} ,{2},{3} }
2. P2 ={ {1,2} , {3} }
3. P3 ={ {2,3} ,{1} }
4. P4 ={ {1,3} ,{2} }
1.1
EXAMPLE
Consider A={a,b,c,d} and let R= { (a,a),(a,b),(b,c),(c,a),(d,c),(c,b) }. Then
R(a)={a,b} , R(b)={c} ,and if A1 ={c,d} ,then R(A1 )={a,b,c} .
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1.2
THEOREM 1
Let R be a relation from A to B , and let A1 and A2 be subsets of A. Then
(a) If A1 ⊆ A2 ,then R(A1 ) ⊆ R(A2 ).
(b)R(A1 ∪ A2 ) = R(A1 ) ∪ R(A2 ) .
(c)R(A1 ∩ A2 ) ⊆ R(A1 ) ∩ R(A2 ).
PROOF:(a) If y ∈ R(A1 ),then x R y for some x ∈ A1 .
Since A1 ⊆ A2 , x ∈ A2 .
Thus , y ∈ R(A2 ) ,which proves part (a).
(b) If y∈R(A1 ∪ A2 ) , then by definition x R y for some x in A1 ∪ A2 .
If x is in A1 ,then since x R y ,we must have y ∈ R(A1 ).
By the same argument, if x is in A2 ,then y ∈ R(A2 ).
In either case y∈ R(A1 ) ∪ R(A2 ).
Thus we have shown that R(A1 ∪ A2 ) ⊆ R(A1 ) ∪ R(A2 ).
Conversly,since A1 ⊆ (A1 ∪ A2 ) ,part (a) tells us that R(A1 ) ⊆ R(A1 ∪ A2 ).
Similarly ,R(A2 ) ⊆ R(A1 ∪ A2 ).
Thus R(A1 ) ∪ R(A2 ) ⊆ R(A1 ∪ A2 ), and there fore part (b) is true.
(c)If y ∈ R(A1 ∩ A2 ),then for some x in A1 ∩ A2 , x R y .
Since x is in both A1 and A2 , it follows that y is in both R(A1 ) and R(A2 );
that is y ∈ R(A1 ) ∩ R(A2 ).
Thus part (c) holds good.
1.3
EXAMPLE
Let A=Z ,R be ” ≤, ”A1 = {0, 1, 2}, and A2 = {9, 13}.
Then R(A1 ) consists of all integers n such that 0 ≤ n, or 1 ≤ n, or 2 ≤ n.
Thus R(A1 ) = {0, 1, 2, ....}.
Similarly ,,R(A2 ) = {9, 10, 11, ......}, so R(A1 ) ∩ R(A2 ) = {9.10.11.....}.
On the other hand A1 ∩ A2 = Φ;
thus R(A1 ∩ A2 ) = Φ.
This shows that the containment in Theorem 1(c) is not always equality.
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1.4
THEOREM 2
Let R and S be relations from A to B. IF R(a)=S(a) for all a in A, then R=S.
Proof
If a R b ,then B ∈ R(a). Therefore ,b∈ S(a) and a S b.
Acompletelysimilarargumentshowsthat,
if aSb, thenaRb.
T husR = S.
1.5
EXAMPLE
We can represent relation between two sets as follows
A= {a1 , a2 , ......, am } and B={b1 , b2 , ......, bn }
Let R is a relation from A to B ,
We represent R by the m x n matrix =[mij ],which is defined by
mij = 1if (ai , bj ) ∈ R
mij = 0if (ai , bj ) ∈
/ R.
The matrix MR is called the matrix of r often MR provides an
easy way to check whether R has a given property.
1.6
EXAMPLE


1 0 0 1
Consider the matrix M =  0 1 1 0 
1 0 1 0
Mij = 1 if (ai , bj ) ∈ R
Since M is 3 x 4 we get A = {a1 , a2 , a3 } and b = {b1 , b2 , b3 , b4 }.
Then (ai , bj ) ∈ R if and only if mij = 1. Thus
R = {(a1 , b1 ), (a1 , b4 ), (a2 , b2 ), (a2 , b3 ), (a3 , b1 ), (a3 , b3 )}.
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