Relative Difference
If A and B are two sets, then the difference between A and B, the relative compliment of B in A, is the set A - B defined by (note that = is just a abuse that
we use to ϵ)
A − B = xϵA : xϵ′ B
This does not necessarily mean B ⊂ A
But when the above condition holds true, then
A − B = A − (A ∩ B)
For the sake of simplicity, we assume a set U , the universe on which we can assume that all sets that we create are subsets of them.
Given a set A, we can define a set which is the absolute compliment of A, which is A’, which also provides that (A’)’ = A.
Also as with all ∅,
∅’ = E and E’ = ∅
Also, its easy to prove that,
A ∩ A′ = ∅
Also,
A ⊂ B if andonlyif , B′ ⊂ A′
Lets proceed to prove the above one. Lets assume the if part, if B′ ⊂ A′ : if xϵA, its self evident that xϵ′ A, and since B′ ⊂ A′ which clearly implies that , x ⊂ B′ which leaves us withxϵAandxϵB. Proceeding with the only if part, A ⊂ B if xϵB′ the clearly, xϵ′ B and since AϵB, its self evident that xϵ′ A, which proves the above one.
Moving forward with the De Morgans Laws,
(A ∪ B)′ = (A′ ∩ B′ )
Lets proceed to prove the above , lets assume xϵ left side then xϵ′ (A ∪ B) which
means, xϵ′ A and xϵ′ B which directly leads to xϵA and xϵB. Proceeding to the
right side, if xϵrightside then xϵ′ A and xϵ′ B which means that x is not present in
either A or B which leads to x being in A or B, hence, (A∪ B)′ = (A′ ∩ B′ ).
Hence, Demorgan Law have been proved. Similiarly, we can also prove that
(A ∩ B)′ = A′ ∩ B′
Principle of duality of Sets
One of the major implication of De morgans law is that, all theorems in Set
theory come in pairs.
So if an inclusion or an equation containing, unions, intersections and compli-
ments of a sub-set E, we replace al l the sub-sets by its compliments, inverse its
unions and intersections and reverse the inclusion , we arrive at a new theorem.
Symettric Difference
A + B = (A − B) ∪ (B − A)
This is the kin of XOR operation http : //en.wikipedia.org/wiki/Exclusivedisjunction
This is commutative and associative.
Why is theory of intersections only applicable to non ∅ sets ?
Lets just assume a set, where xϵX : xϵof ∅
Lets try to prove the other part, If this does not hold true, then ∅ must contain
a set X such that xϵX ′ , which of course is ridiculous, which proves the point.
To clear this, we can define a sub set in the Universe U, such that, ϱ(All subsets
of universe)
xϵU |xϵX f oreveryX inϱ
