Symmetric Difference Operator and tricks

if P and Q are set then Symmetric difference Δ\Delta is given by

P \Delta Q = (P \cup Q) - (P \cap Q)

for Simplicity :

\cap = .            // product operator

\cup = +       //  Sum operator

and 

A- B = A. B^{c} 

Now 

P \Delta Q = (P \cup Q) - (P \cap Q)

               = (P+ Q) -(PQ)

              = (P+Q).(PQ)^{c}

              = (P+Q).(P^{c} + Q^{c})

              = PQ^{^{c}} + P^{^{c}}Q

If you remember it is the formula for XOR 

The symmetric difference Δ\Delta is nothing but XOR operator.

for convenient we can apply all the property of XOR where Symmetric Difference Operator is used. 

 

Example:-

Let E , F and G be finite sets and  

X = (E \cap F) - (F \cap G)

Y = (E - (E \cap G)) - (E -F)

 

What is the relation between X and Y  ?

X = EF . (FG)^{c}

        = EF . (F^{c} + G^{c})

       = EFG^{c}

Similarly 

Y = EFG^{c} 

X=Y = EFG^{c}

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