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.
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