Q4
Let U=the set of the English alphabets, A={x l x is a vowel} ,B{y l y is a consonant}, verify De Morgan's Laws for these sets.
SOLUTION:
Here U={a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z}
Vowels=a,e,i,o,u so A={a,e,i,o,u}
Constant=b,c,d,f,g,h,j,k,l,m,n,p,q,r,s,t,v,w,x,y,z,so
B={b,c,d,f,g,h,j,k,l,m,n,p,q,r,s,t,v,w,x,y,z}
(i) (AUB)c=Ac∩Bc
L.H.S=(AUB)c
=U-(AUB)
={a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z}-({a,e,i,o,u}U{{b,c,d,f,g,h,j,k,l,m,n,p,q,r,s,t,v,w,x,y,z})
={a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z}-{a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z}
={ }
Now taking R.H.S
R.H.S=Ac∩Bc
=(U-A)∩(U-B)
=({a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z}-{a,e,i,o,u})∩({a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z}-{b,c,d,f,g,h,j,k,l,m,n,p,q,r,s,t,v,w,x,y,z})
={b,c,d,f,g,h,j,k,l,m,n,p,q,r,s,t,v,w,x,y,z}∩{a,e,i,o,u}
={ }
Hence L.H.S=R.H.S
(AUB)c=Ac∩Bc
(ii) (A∩B)c=AcUBc
L.H.S=(A∩B)c
=U-(A∩B)
={a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z}-({a,e,i,o,u}∩{b,c,d,f,g,h,j,k,l,m,n,p,q,r,s,t,v,w,x,y,z})
={a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z}-{ }
={a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z}
Now taking R.H.S
R.H.S=AcUBc
=(U-A)U(U-B)
=({a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z}-{a,e,i,o,u})U({a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z}-{b,c,d,f,g,h,j,k,l,m,n,p,q,r,s,t,v,w,x,y,z})
={b,c,d,f,g,h,j,k,l,m,n,p,q,r,s,t,v,w,x,y,z}U{a,e,i,o,u}
={a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z}
Hence L.H.S=R.H.S
(A∩B)c=AcUBc