Fsc Part 1 Mathematics (Complete Solution)

# Q2 Verify the properties for the sets a,b and c given below

(i) Associativity of Union

(ii) Associativity of intersection

(iii) Distributivity of Union over intersection

(iv) Distributivity if intersection over union

(a) A={1,2,3,4}, B{3,4,5,6,7,8}, C={5,6,7,9,10}

SOLUTION:

(i) Associativity of Union

(AUB)UC=AU(BUC)

L.H.S=(AUB)UC

=({1,2,3,4}U{3,4,5,6,7,8})U{5,6,7,9,10}

={1,2,3,4,5,6,7,8}U{5,6,7,9,10}

={1,2,3,4,5,6,7,8,9,10}

R.H.S=AU(BUC)

={1,2,3,4}U({3,4,5,6,7,8}U{5,6,7,9,10})

={1,2,3,4}U{3,4,5,6,7,8,9,10}

={1,2,3,4,5,6,7,8,9,10}

L.H.S=R.H.S

(ii) Associativity of intersection

(A∩B)∩C=A∩(B∩C)

L.H.S=(A∩B)∩C

=({1,2,3,4}∩{3,4,5,6,7,8})∩{5,6,7,9,10}

={3,4}∩{5,6,7,9,10}

={  }

R.H.S=A∩(B∩C)

={1,2,3,4}∩({3,4,5,6,7,8}∩{5,6,7,9,10})

={1,2,3,4}∩{5,6,7}

={  }

L.H.S=R.H.S

(iii) Distributivity of union over intersection.

AU(B∩C)=(AUB)∩(AUC)

L.H.S=AU(B∩C)

={1,2,3,4}U({3,4,5,6,7,8}∩{5,6,7,9,10})

={1,2,3,4}U{5,6,7}

={1,2,3,4,5,6,7}

R.H.S=(AUB)∩(AUC)

=({1,2,3,4}U{3,4,5,6,7,8})∩({1,2,3,4}U{5,6,7,9,10})

={1,2,3,4,5,6,7,8}∩{1,2,3,4,5,6,7,9,10}

={1,2,3,4,5,6,7}

L.H.S=R.H.S

(iv) Distributivity of intersection over union.

A∩(BUC)=(A∩B)U(A∩C)

L.H.S=A∩(BUC)

={1,2,3,4}∩({3,4,5,6,7,8}U{5,6,7,9,10})

={1,2,3,4}∩{3,4,5,6,7,8,9,10}

={3,4}

R.H.S=(A∩B)U(A∩C)

=({1,2,3,4}∩{3,4,5,6,7,8})U({1,2,3,4}∩{5,6,7,9,10})

={3,4}U{  }

={3,4}

L.H.S=R.H.S

(b) A=Φ, B={0}, C={0,1,2}

SOLUTION:

(i) Associativity of union.

(AUB)UC=AU(BUC)

L.H.S=(AUB)UC

=( ΦU{0})U{0,1,2}

={0}U{0,1,2}

={0,1,2}

R.H.S=AU(BUC)

= ΦU({0}U{0,1,2})

=ΦU{0,1,2}

={0,1,2}

L.H.S=R.H.S

(ii) Associativity of intersection.

(A∩B)∩C=A∩(B∩C)

L.H.S=(A∩B)∩C

=( Φ∩{0})∩{0,1,2}

=Φ∩{0,1,2}

R.H.S=A∩(B∩C)

= Φ∩({0}∩{0,1,2})

=Φ∩{0}

L.H.S=R.H.S

(iii) Distributivity of union over intersection

AU(B∩C)=(AUB)∩(AUC)

L.H.S=AU(B∩C)

= ΦU({0}∩{0,1,2})

= ΦU{0}

={0}

R.H.S=(AUB)∩(AUC)

=(ΦU{0})∩(ΦU{0,1,2})

={0}∩{0,1,2}

={0}

L.H.S=R.H.S

(iv) Distributivity of intersection over union.

A∩(BUC)=(A∩B)U(A∩C)

L.H.S=A∩(BUC)

= Φ∩({0}U{0,1,2})

=Φ∩{0,1,2}

R.H.S=(A∩B)U(A∩C)

=(Φ∩{0})U(Φ∩{0,1,2})

=ΦUΦ

L.H.S=R.H.S

(c) N,Z,Q

SOLUTION:

(i) Associativity of union.

(NUZ)UQ=NU(ZUQ)

L.H.S=(NUZ)UQ

=ZUQ

=Q

R.H.S=NU(ZUQ)

=NUQ

=Q

L.H.S=R.H.S

(ii) Associativity of intersection.

(N∩Z)∩Q=N∩(Z∩Q)

L.H.S=(N∩Z)∩Q

=N∩Q

=N

R.H.S=N∩(Z∩Q)

=N∩Z

=N

L.H.S=R.H.S

(iii) Distributivity of union over intersection

NU(Z∩Q)=(NUZ)∩(NUQ)

L.H.S=NU(Z∩Q)

=NUZ

=Z

R.H.S=(NUZ)∩(NUQ)

=Z∩Q

=Z

(iv) Distributivity of intersection over union.

N∩(ZUQ)=(N∩Z)U(N∩Q)

L.H.S=N∩(ZUQ)

=N∩Q

=N

R.H.S=(N∩Z)U(N∩Q)

=NUN

=N

L.H.S=R.H.S