SYMMETRIC DIFFERENCE OF TWO SETS
About "Symmetric difference of two sets"
Symmetric difference of two sets :
Symmetric difference is one of the important operations on sets
Let us discuss this operation in detail.
Let X and Y be two sets.
Now, we can define the following new set.
X Δ Y = (X \ Y) u (Y \ X)
X Δ Y is read as "X symmetric difference Y"
Now that X Δ Y contains all elements in X u Y which are not in X n Y and the figure given below illustrates this. .
Example :
Let A = { 1, 3, 5, 6 }, B = { 0, 5, 6, 7 }, find A Δ B.
Solution :
A \ B = { 1, 3, 5, 6 } \ { 0, 5, 6, 7 } = { 1, 3 }
B \ A = { 0, 5, 6, 7 } \ { 1, 3, 5, 6 } = { 0, 7 }
A Δ B = (A \ B) u (B \ A)
A Δ B = { 1, 3 } u { 0, 7 }
A Δ B = { 1, 3, 0, 7 }
Apart from the operation symmetric difference on sets, we have some other important operations on sets.
When two or more sets are combined together to form another set under some given conditions, then these operations on sets are carried out.
Let us discuss the other important operations here:
The other important operations on sets are.
1. Union
2. Intersection
3. Set difference
4. Complement
5. Disjoint sets
Let us discuss the above operations in detail one by one.
Union
Let X and Y be two sets.
Now, we can define the following new set.
X u Y = {z | z ∈ X or z ∈ Y}
(That is, z may be in X or in Y or in both X and Y)
X u Y is read as "X union Y"
Now that X u Y contains all the elements of X and all the elements of Y and the figure given below illustrates this.
It is clear that X ⊆ X u Y and also Y ⊆ X u Y
Intersection
Let X and Y be two sets.
Now, we can define the following new set.
X n Y = {z | z ∈ X and z ∈ Y}
(That is z must be in both X and Y)
X n Y is read as "X intersection Y"
Now that X n Y contains only those elements which belong to both X and Y and the figure given below illustrates this.
It is trivial that that X n Y ⊆ X and also X n Y ⊆ Y
Set difference
Let X and Y be two sets.
Now, we can define the following new set.
X \ Y = {z | z ∈ X but z ∉ Y}
(That is z must be in X and must not be in Y)
X \ Y is read as "X difference Y"
Now that X \ Y contains only elements of X which are not in Y and the figure given below illustrates this.
Some authors use A - B for A \ B. We shall use the notation A \ B which is widely used in mathematics for set difference.
Complement
If X ⊆ U, where U is a universal set, then U \ X is called the compliment of X with respect to U. If underlying universal set is fixed, then we denote U \ X by X' and it is called compliment of X.
X' = U \ X
The difference set set A \ B can also be viewed as the compliment of B with respect to A.
Disjoint sets
Two sets X and Y are said to be disjoint if they do not have any common element. That is, X and Y are disjoint if
X n Y = ᵩ
It is clear that n(A u B) = n(A) + n(B), if A and B are disjoint finite set.
After having gone through the stuff given above, we hope that the students would have understood "Symmetric difference of two sets".
Apart from the stuff given above, if you want to know more about "Symmetric difference of two sets", please click here
Apart from the stuff "symmetric difference of two sets", if you need any other stuff in math, please use our google custom search here.
WORD PROBLEMS
HCF and LCM word problems
Word problems on simple equations
Word problems on linear equations
Word problems on quadratic equations
Area and perimeter word problems
Word problems on direct variation and inverse variation
Word problems on comparing rates
Converting customary units word problems
Converting metric units word problems
Word problems on simple interest
Word problems on compound interest
Word problems on types of angles
Complementary and supplementary angles word problems
Markup and markdown word problems
Word problems on mixed fractrions
One step equation word problems
Linear inequalities word problems
Ratio and proportion word problems
Word problems on sets and venn diagrams
Pythagorean theorem word problems
Percent of a number word problems
Word problems on constant speed
Word problems on average speed
Word problems on sum of the angles of a triangle is 180 degree
OTHER TOPICS
Time, speed and distance shortcuts
Ratio and proportion shortcuts
Domain and range of rational functions
Domain and range of rational functions with holes
Graphing rational functions with holes
Converting repeating decimals in to fractions
Decimal representation of rational numbers
Finding square root using long division
L.C.M method to solve time and work problems
Translating the word problems in to algebraic expressions
Remainder when 2 power 256 is divided by 17
Remainder when 17 power 23 is divided by 16
Sum of all three digit numbers divisible by 6
Sum of all three digit numbers divisible by 7
Sum of all three digit numbers divisible by 8
Sum of all three digit numbers formed using 1, 3, 4
Sum of all three four digit numbers formed with non zero digits
Sum of all three four digit numbers formed using 0, 1, 2, 3
Sum of all three four digit numbers formed using 1, 2, 5, 6
