properties of symmetric difference

properties of symmetric difference

Recall that the

symmetric difference

of two sets

A

,

B

is the set

A

∪

B

-

(

A

∩

B

)

. In this entry, we list and prove some of the basic

properties

of

△

.

1.

(

commutativity

of

△

)

A

⁢

△

⁢

B

=

B

⁢

△

⁢

A

, because

∪

and

∩

are

commutative

.

2.

If

A

⊆

B

, then

A

⁢

△

⁢

B

=

B

-

A

, because

A

∪

B

=

B

and

A

∩

B

=

A

.

3.

A

⁢

△

⁢

∅

=

A

, because

∅

⊆

A

, and

A

-

∅

=

A

.

4.

A

⁢

△

⁢

A

=

∅

, because

A

⊆

A

and

A

-

A

=

∅

.

5.

A

⁢

△

⁢

B

=

(

A

-

B

)

∪

(

B

-

A

)

(hence the name symmetric difference).

Proof.

A

⁢

△

⁢

B

=

(

A

∪

B

)

-

(

A

∩

B

)

=

(

A

∪

B

)

∩

(

A

∩

B

)

′

=

(

A

∪

B

)

∩

(

A

′

∪

B

′

)

=

(

(

A

∪

B

)

∩

A

′

)

∪

(

(

A

∪

B

)

∩

B

′

)

=

(

B

∩

A

′

)

∪

(

A

∩

B

′

)

=

(

B

-

A

)

∪

(

A

-

B

)

.
∎

6.

A

′

⁢

△

⁢

B

′

=

A

⁢

△

⁢

B

, because

A

′

⁢

△

⁢

B

′

=

(

A

′

-

B

′

)

∪

(

B

′

-

A

′

)

=

(

A

′

∩

B

)

∪

(

B

′

∩

A

)

=

(

B

-

A

)

∩

(

A

-

B

)

=

A

⁢

△

⁢

B

.

7.

(

distributivity

of

∩

over

△

)

A

∩

(

B

⁢

△

⁢

C

)

=

(

A

∩

B

)

⁢

△

⁢

(

A

∩

C

)

.

Proof.

A

∩

(

B

⁢

△

⁢

C

)

=

A

∩

(

(

B

∪

C

)

-

(

B

∩

C

)

)

, which is

(

A

∩

(

B

∪

C

)

)

-

(

A

∩

(

B

∩

C

)

)

, one of the

properties of set difference

(see proof here (

http://planetmath.org/PropertiesOfSetDifference

)). This in turns is equal to

(

(

A

∩

B

)

∪

(

A

∩

C

)

)

-

(

(

A

∩

B

)

∩

(

A

∩

C

)

)

=

(

A

∩

B

)

⁢

△

⁢

(

A

∩

C

)

.
∎

8.

(

associativity

of

△

)

(

A

⁢

△

⁢

B

)

⁢

△

⁢

C

=

A

⁢

△

⁢

(

B

⁢

△

⁢

C

)

.

Proof.

Let

U

be a set containing

A

,

B

,

C

as subsets (take

U

=

A

∪

B

∪

C

if necessary). For a given

B

, let

f

:

P

⁢

(

U

)

×

P

⁢

(

U

)

→

P

⁢

(

U

)

be a

function

defined by

f

⁢

(

A

,

C

)

=

(

A

⁢

△

⁢

B

)

⁢

△

⁢

C

. Associativity of

△

is then then same as showing that

f

⁢

(

A

,

C

)

=

f

⁢

(

C

,

A

)

, since

A

⁢

△

⁢

(

B

⁢

△

⁢

C

)

=

(

B

⁢

△

⁢

C

)

⁢

△

⁢

A

=

(

C

⁢

△

⁢

B

)

⁢

△

⁢

A

.

By expanding

f

⁢

(

A

,

C

)

, we have

(

A

⁢

△

⁢

B

)

⁢

△

⁢

C

=

(

(

A

⁢

△

⁢

B

)

-

C

)

∪

(

C

-

(

A

⁢

△

⁢

B

)

)

=

(

(

(

A

-

B

)

∪

(

B

-

A

)

)

∩

C

′

)

∪

(

C

-

(

(

A

∪

B

)

-

(

A

∩

B

)

)

)

=

(

(

(

A

∩

B

′

)

∪

(

B

∩

A

′

)

)

∩

C

′

)

∪

(

(

C

∩

A

∩

B

)

∪

(

C

-

(

A

∪

B

)

)

=

(

(

A

∩

B

′

∩

C

′

)

∪

(

B

∩

A

′

∩

C

′

)

)

∪

(

(

C

∩

A

∩

B

)

∪

(

C

∩

A

′

∩

B

′

)

)

=

(

B

∩

A

′

∩

C

′

)

∪

(

B

∩

A

∩

C

)

∪

(

B

′

∩

A

∩

C

′

)

∪

(

B

′

∩

A

′

∩

C

)

.

It is now easy to see that the last

expression

does not change if one exchanges

A

and

C

. Hence,

f

⁢

(

A

,

C

)

=

f

⁢

(

C

,

A

)

and this shows that

△

is associative.
∎

Remark

. All of the properties of

△

on sets can be generalized to

△

(

http://planetmath.org/DerivedBooleanOperations

) on

Boolean algebras

.

Title

properties of symmetric difference

Canonical

name

PropertiesOfSymmetricDifference

Date of creation

2013-03-22 14:36:56

Last modified on

2013-03-22 14:36:56

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

14

Author

CWoo (3771)

Entry type

Derivation

Classification

msc 03E20

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