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=A^c∩B^c

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=A^c∩B^c

=(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=A^c∩B^c

(ii)  (A∩B)^c=A^cUB^c

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=A^cUB^c

=(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=A^cUB^c