# Operations on Sets

Notations:-

∈ → belongs to

∉ → does not belongs to

⊆ → is a subset of

⊇ → is a superset of

⊂ → is a proper subset of

∀ → for all/ for every/ for each

∃ → there exists (at least one)

(i) Union of sets: Given ‘n’ sets n  2, then the set formed by taking all the elements of each of the n-sets as its element as its elements is called union of n-sets

The union of two sets A, B is denoted by A ∪ B and defined as A ∪ B = {x| x ϵ A V x ϵ B} or  $$\bigcup\limits_{i=1}^{n}{{{A}_{i}}}$$.

⇒ $$\bigcup\limits_{i=1}^{n}{{{A}_{i}}}$$ = {x| x ϵ at least one of sets A1, A2, — An}Note: – Every set is a subset of itself

(ii) Null set is subset of every set:

A ⊆ A ∪ B, B ⊆ A ∪ B

If A, B are infinite disjoint sets |A ∪ B| = | A | + | B |

∩ | A ∪ B | = ∩ (A) + ∩ (B)In general, N sets A1, A2, …, An are finite pair wise defined as $$\left| \bigcup\limits_{i=1}^{n}{{{A}_{i}}} \right|=\sum\limits_{i=1}^{n}{\left| Ai \right|}$$.

Properties of union of sets:

⇒ A ∪ A = A … {Idempotent law under union}

⇒ A ∪ B = B ∪ A … {Commutative}

⇒ A ∪ B ∪ C = (A ∪B) ∪ C = A ∪ (B ∪ C) … (associative)