CLASS-11SOME IMPORTANT THEOREM ON SUBSETS

Some Theorems On Subset -

Theorem.1)   Every set is a subset of itself.

Theorem.2) The empty set is a subset of all Sets

For example, if P = {1, 3, 5, 7}, Q = {1, 3, 5, 7}, then Q ⊆ P and ϕ ⊆ P

If, P is a subset of empty set, if P ⊆ ϕ, then P = ϕ,

P itself is the empty set. Thus ϕ ⊆ ϕ

Theorem.3) If P ⊆ Q, and Q ⊆ R, then P ⊆ R

Proof.  Let, x ∈ P, then since P ⊆ Q, x ∈ Q.

Now, x ∈ Q

=>   x ∈ R because Q ⊆ R

x ∈ P, and x ∈ R

so, P ⊆ R

Theorem.4)  For any two sets P & Q, if P ⊆ Q, and Q ⊆ P, then P = Q and vice versa

Proof.-   Let, x ∈ P, then x ∈ Q since P ⊆ Q …………..(i)

Also, x ∈ Q, then x ∈ P since Q ⊆ P …………………..(ii)

Conversely, let P ⊆ Q, then

x ∈ P  =>  x ∈ Q, since P = Q,

So, P ⊆ Q

Similarly,  x ∈ Q  => x ∈ P  (since P = Q),

So,  Q ⊆ P

For example, let P = {Richard, Pollard, John}, then all the possible subsets of P are

Q = {Richard, Pollard, John}, R = {Richard, Pollard}, S = {Richard, John}, T = {Pollard, John}, U = {Richard}, V = {Pollard}, W = {John}, X = ϕ

The sets Q and X above are considered as subsets of P, Q ⊆ P and X ⊆ P but sets R, S, T, U, V, and W are considered proper subsets of P, R  ⊂  P, S ⊂ P, T ⊂ P, U ⊂ P, V ⊂ P, W ⊂ P.