Question

Given L = {1, 2, 3, 4}, M = {3, 4, 5, 6} and N = {1, 3, 5}. Verify that L – (M ∪ N) = (L – M) ∩ (L – N)

Answer 1

The union of two sets is a set containing all elements that are in both sets.

For example: {1, 2, 3} ∪ {2, 4} = {1, 2, 3, 4}

The difference (subtraction) is defined as: The set A – B consists of elements that are in A but not in B.

For example: if A = {1, 2, 3} and B = {3, 5}, then A−B = {1, 2}

The intersection of two sets A and B, consists of all elements that are both in A and B.

For example: {1, 2} ∩ {2, 3} = {2}

Therefore,

M = {3, 4, 5, 6}, N = {1, 3, 5} 

⇒ M ∪ N = {1, 3, 4, 5, 6}

L = {1, 2, 3, 4} and M ∪ N = {1, 3, 4, 5, 6}

⇒ L – (M ∪ N) = {2}  ......(1)

L = {1, 2, 3, 4} and M = {3, 4, 5, 6} 

⇒ L – M = {1, 2}

L = {1, 2, 3, 4} and N = {1, 3, 5} 

⇒ L – N = {2, 4}

L – M = {1, 2} and L – N = {2, 4}

⇒ (L – M) ∩ (L – N) = {2}  .......(2)

Clearly, from (1) and (2)

L – (M ∪ N) = (L – M) ∩ (L – N)

Hence verified