Important points on Set


Subset

A⊆ B is defined as x∈  A⇒ x ∈  B.

If x ∈ A ⇒ x∈ B then A⊆B.

Equal set

If A⊆B and B⊆A then A = B.

Union of sets

A∪B = {x: x ∈ A or x∈ B}

x∈ A∪B ⇒ x ∈ A or x∈ B.

but x∉ A∪B ⇒ x∉ A and x∉ B.

Intersection of sets

A∩B = {x: x∈ A and x∈ B}

x∈ A∩B ⇒ x ∈ A and x∈ B.

but x∉ A∩B ⇒ x∉ A or x∉ B.

Difference of sets

A – B = {x: x∈ A and x∉ B}

B – A = {x: x∈ B and x ∉ A}

Complement of set

A̅ = { x: x ∈ U and x∉ A}

= {x : x ∉ A}

* x∈ A̅ ⇒ x∉ A

* x∈ A ⇒ x∉ A̅


Symmetric Difference

*Union of A – B and B – A

*Denoted by A∆B

∴ A∆B = (A – B)∪(B – A)

            = {x: x ∈ A – B or x ∈ B – A}

∴ x ∈ A∆B ⇒ x∈ A – B or x∈ B – A

∴ x ∈ A∆B ⇒ x∉ A∩B

* x ∈ A and x∉ A ⇒ x ∈ Ï•  

* x∈ A or x∈ A̅ ⇒ x∈ U