Symmetric Difference of Two Sets And Their Algebraic Properties

Some use sets consciously and some unconsciously. Here, we delve deep into an interesting part of set language, i.e. the symmetric difference of two sets. Suppose you have a collection of novels and let us name your collection as set A. Your friend’s collection is set B. 

Set A = {The Great Gatsby, Pride and Prejudice, Wuthering Heights, Oliver Twist} 

Set B = {Oliver Twist, The Tale of Two Cities, Hamlet, Ulysses, The Great Gatsby, Wuthering Heights} 

Now the symmetric difference of these two sets A and B is {Pride and Prejudice, The Tale of Two Cities, Hamlet} 

Symmetric Difference Of Two Sets Meaning 

From the above example, it is clear that the symmetric difference of two sets A and B contains all the elements of Sets A and B but their intersection. The intersection of two sets is the common element found in the two sets. Thus, the lack of connection or disjunction between the two sets. 

So, the symmetric difference can be defined as the set of all the elements contained in the sets minus their common elements. The symmetric difference is denoted by the symbol – ‘Δ’ (delta). 

 Type 1 – P △ Q = (P U Q) – (Q ∩ P) 

Similarly, Q △ P = (Q U P) – (P ∩ Q)  

Type 2 – P △ Q = (P U Q) – (P ∩ Q) 

Thus, there are two different forms to represent the symmetric difference. 

Type 1 – Examples

For example, Set C {1, 2, 3, 4, 9, 6, 11, 13, 14, 15, 16, 17, 20}

Set D {1, 2, 5, 6, 7, 11, 13, 19, 21, 23, 25, 27, 29}

C △ D = (C U D) – (D ∩ C) 

Now, (C U D) = {1, 2, 3, 4, 9, 6, 11, 13, 14, 15, 16, 17, 20, 5, 7, 19, 21, 23, 25, 27, 29}

(D ∩ C) = {1, 2, 6, 11, 13} 

Now, find the difference between sets C and D. 

So, (C U D) – (D ∩ C) is {3, 4, 9, 14, 15, 16, 17, 20, 5, 7, 19, 21, 23, 25, 27, 29}

Therefore, C △ D = {3, 4, 9, 14, 15, 16, 17, 20, 5, 7, 19, 21, 23, 25, 27, 29}

Likewise for D △ C = (D U C) – (C ∩ D) 

(D U C) = {1, 2, 3, 4, 9, 6, 11, 13, 14, 15, 16, 17, 20, 5, 7, 19, 21, 23, 25, 27, 29}

(C ∩ D) = {1, 2, 6, 11, 13}

So, the difference of (D U C) – (C ∩ D) is {3, 4, 9, 14, 15, 16, 17, 20, 5, 7, 19, 21, 23, 25, 27, 29}

Therefore, D △ C is {3, 4, 9, 14, 15, 16, 17, 20, 5, 7, 19, 21, 23, 25, 27, 29}

Type – 2 Examples

Set A = {1, 2, 3, 4, 5, 6, 7} 

Set B = {9, 5, 7, 4, 3, 1, 2}

A △ B = (A U B) – (A ∩ B)

(A U B) = {1, 2, 3, 4, 5, 6, 7, 9}

(A ∩ B) = {1, 2, 3, 4, 5, 7}

(A U B) – (A ∩ B) = {6, 9} 

Algebraic Properties 

Algebraic properties of sets make evaluating expressions effortless. These expressions include union, intersection, and complement of sets. The basic properties are commutative property, associative property, and distributive property. 

However, as only two sets are involved in the symmetric difference of two sets, associative property and distributive property do not apply here. 

• Commutative Property for symmetric difference – (A △ B) = (B △ A) 

Set A = {1, 2, 3, 4, 5, 6, 7} 

Set B = {9, 5, 7, 4, 3, 1, 2}

L.H.S:

(A △ B) = (A – B) U (B – A) 

(A – B) = {6}

(B – A) = {9, 2}

(A – B) U (B – A) = {6, 9, 2} ————— (1)

 R.H.S:  

Now, (B △ A)

(B △ A) = (B – A) U (A – B)

(B – A) = {9, 2}

(A – B) = {6}

(B – A) U (A – B) = {6, 9, 2}————— (2)

L.H.S = R.H.S 

(A – B) U (B – A) = {6, 9, 2} = (B – A) U (A – B) = {6, 9, 2}

So, (A △ B) = (B △ A) 

Hence, proved.

Conclusion 

The above given was the symmetric difference of two sets notes. We also saw about the symmetric difference of two sets meaning, examples, fundamental application of algebraic properties in sets and their examples. There are several other algebraic properties related to set theory that are worth knowing about like the principle of duality, laws of union and intersection, De Morgan’s Law, etc. Learning such topics help in a better and deeper understanding of set-related topics. Further symmetric difference of two sets and such mathematical concepts are fascinating as well. Only commutative property is applicable of all the basic properties, as in this topic, we deal with two sets.