A plane closed figure used to represent sets and the relationship between different sets is called Venn Diagram in mathematics. Venn diagram consists of universal sets ‘U’ and related sets under consideration. We denote a universal set by a rectangle and the related sets are represented by circles or oval shapes. Furthermore, the closed figures of sets are kept inside the rectangle. If sets are folded in each other then they are said to be overlapping otherwise they are called disjoint sets in mathematics.
Example:
What are the disjoint and overlapping sets in the Venn diagram?
Disjoint Sets:
Disjoint sets are the sets that have no common elements between them or the intersection of these sets is an empty set. In other words, If M and N are any two sets then they are said to be disjoint if there is no common element between M and N and M⋂N = ɸ
Example:
Let M = {r, s, t,} and N = {a, x, y,} then
M⋂N = {r, s, t,} ⋂ {a, x, y,}
= { ɸ }
Thus, M⋂N = ɸ
So, M and N are the disjoint sets in mathematics.
In the Venn diagram:
Overlapping Sets:
If there is at least one common element between any two sets then they are called overlapping sets in mathematics. In addition, the intersection of these two sets is also not an empty set.
i.e For any two sets F and G are said to be overlapping sets if there is at least one common element between F and G and F⋂G ≠ ɸ
Example:
Suppose that F = {3. 6. 9. 12} and G = {4, 8, 12, 16} then
F⋂G = {3, 6, 9, 12} ⋂ {4, 8, 12, 16}
= {12}
≠ ɸ
Hence, F⋂G ≠ ɸ which shows that F and G are overlapping sets in mathematics.
In Venn Diagram:
Operations on sets in math by Venn Diagram:
The operations on stets in mathematics are represented by the shaded region in the Venn diagram. Now, we are going to study some operations on sets by using a Venn diagram.
Union of Sets:
Union of any two sets is a set that contains the elements of both sets and is denoted by a symbol ‘U’. Now, we are going to apply an operation of the union to show the Venn diagram on different combinations of sets.
For Subsets:
If Any set is a subset of another set then it is considered as a union of these two sets and can be represented by a Venn diagram.
Example:
If R = {e, f, g} and S = {a, e, f, g h} then R is a subset of S and also
RUS = {a, e, f, g, h } which can be represented by the Venn diagram as:
For Overlapping sets
In overlapping sets, only a few elements will be common in any two sets and can be represented by the Venn diagram.
Example:
Let U = {p, q, r, s} and V = {r, s, t, u}
Here, U and V have some common elements then UUV is represented by the Venn diagram and shaded as:
For Disjoint Sets:
Two sets with no common elements are said to be disjoint and are represented by the Venn diagram.
Example:
Consider that K = {apple, orange, pear} and L = {triangle, rectangle, circle} that have no common elements then KUL can be represented by Venn diagram as:
Intersection Of sets:
The intersection of any two sets is a set that contains only common elements between these two sets. The intersection of any two sets is denoted by a symbol ‘∩’. Now, we are going to draw a Venn diagram by applying an operation of intersection on different combinations of sets.
For subsets:
If T is a subset of S but S is not a subset of T then, we can say that there is an operation between these two sets i.e S ∩ T.
For Example: If T = {p, q, r} and S = {l, m, n, o, p, q, r}
Here T ⊆ S and
T∩S = {p, q, r} then it can be shown by Venn diagram as:
For Disjoint Sets:
For any two sets that have no common elements between them and their intersection must be an empty set is called a disjoint set.
Example:
Suppose H = {pencil, bag, book} and I = {chocolate, cake, pastry} then H and I are disjoint sets and can be represented by Venn diagram as:
For Overlapping sets:
Two sets that have at least one common element between them and their intersection must a non-empty set is called overlapping set.
Example:
W = {5, 10, 15, 20} and X = {3, 6, 9, 12, 15} that are overlapping sets and can be represented by venn diagram as:
For Difference of two sets:
The difference between any two sets is a set that contains only those elements of the first set that do not belong to the second set. The operation of difference of two sets can be applied to the different combinations of sets in mathematics and also shown in the Venn diagram that we are going to study below:
Subsets:
For any two sets, If all the elements of the first set belong to the second set then their difference is an empty set and can be represented by Venn diagram i.e.
If P = {red, green, yellow} and Q = {yellow, green, orange, red}
Here, P is a subset of Q and P-Q is an empty set and can be represented by the Venn diagram as:
For disjoint Sets:
If any two sets are disjoint then their difference is not an empty set and is represented by the Venn diagram.
If J = {u, v, w} and K = {l, m, n} that have no common elements between them and also disjoint sets then their difference is not an empty set i.e.
J-K = {u, v, w} and can be shown in the Venn diagram as:
For Overlapping Sets:
When any two sets have some common elements and are also called overlapping then the difference is not an empty set and can be represented by a Venn diagram.
Let O = {Facebook, youtube, instagram} and N = {Twitter, TikTok, youtube} here O and N are overlapping sets and O-N is not an empty set and can be represented by the Venn diagram as:
Complement of a Set:
Then complement of a set Y with a universal set U can be obtained by
Y’ = U-Y.
If U = {parrot, pigeon, dove, crow, peacock} and Y = {peacock, crow} then Y’ = {parrot, dove pigeon} and shown by Venn diagram as:
Practice Questions For the Representation of sets by Venn diagram
Question No. 1:
Find out the given operations on sets by using Venn diagram If
U = {a, b, c, d, e}, F = {a, b, c} and G = {b, d, e},
(i)F’
(ii)G’
(iii)F U G
(iv)F ∩ G
(v) F-G
(vi) G-F
(vii) F U G = G U F
(viii) F ∩ G = G ∩ F
Question No.2:
By using representation of sets in mathematics for the following sets U= {1, 2, 3, 10}, P={1,4, 8, 9, 10} and R = {2, 3, 4, 7, 10} show that by using Venn diagram
(i) P - R ≠ R - P
(ii) P ∩ R = R ∩ P
(iii) P U R = R U P
(iv) P’ ≠ R’