1
Discrete Mathematics Structure
1- Prove the following:
a. AU(BI∩C)=(AUBI)∩(AUC)
b. A-B=A∩BI
2- Under what conditions can AUB=A∩B?
3- Let A={1,2,3,4,5} and R={(1,2),(1,1),(2,1),(2,2),(3,3),(4,4),(4,5),(5,4),(5,5)} be equivalence relation on A. Determine the partition corresponding to R-1 if it equivalence relation.
4- Let A=B=C=R and consider the function f:A→B and g:B→C define by f(a)=2a+1, g(b)=b/3, verify (gof)-1=f-1og-1
5- Show that 1.2+2.3+3.4+…………..+n(n+1)=n(n+1)(n+2)/3, n≥1 by mathematical induction.
2
1- Consider the POSET A= ({1,2,3,4,6,9,12,18,36}/). Find the greatest lower bound and least upper bound of the sets {4, 12, 18} and {3, 9, 12}.
2- Prove that power set P of a set S is a Lattice under the operation ∩ and U.
3- Using the Laws show that n(np^q)^(pvq)==p.
4- Obtained DNF of (a) p→q (b) qv(pvnq)
3
1- Determine whether the set P(X) of all subsets of X, under composition * defined by-
A*B=AUB, for all A⊆X and B⊆X forms a group.
2- Prove that the set of cube roots of unity is an Abelian finite group with respect to multiplication.
3- Find the order of each element in the following multiplicative group G
G={a,a2,a3,a4,a5,a6=e}
4- Prove that the ring of rational no. (Q,+,*) is a field.
5- Let A={a,b,c}. Then prove (P(A),R) is a POSET where R is rational “is subset of”. Find LUB and GLB of {{a},{b},{c}}.
