Intersection of the sets

For finding the intersection of two sets, we normally choose those elements which are common in both the sets. If there are three sets, then we choose those elements which are common in all three sets. Therefore, if there are n numbers of sets, then we select only those elements which are common in all the n sets. In this way, we find the intersection of sets. The symbol of intersection of set is denoted as ‘∩’. If A and B are two sets, then the intersection is denoted with the symbol A ∩ B and is pronounced as A intersection B and mathematically, we can represent it as

A∩B={x:x∈A∧x∈B}

Example: A = {1,2,3,4,5}

B = {2,3,7}

A ∩ B = {2,3}

Question3:

If A = {1, 2, 3, 4, 5, 6, 7}, B = {2, 4, 6, 8, 10} and C = {4, 6, 7, 8, 9, 10, 11}, then find A ∩ B and A ∩ B ∩ C.

Solution:

The given sets are

A = {1, 2, 3, 4, 5, 6, 7}

B = {2, 4, 6, 8, 10}

C = {4, 6, 7, 8, 9, 10, 11}

First, we need to find A ∩ B. Then, we need to treat A ∩ B as a single set.

For A ∩ B, we choose those elements which are common in sets A and B.

Therefore, A ∩ B = {2, 4, 6}

For (A ∩ B) ∩ C, we choose those elements which are common in sets A ∩ B and C.

Therefore, (A ∩ B) ∩ C = {4, 6}

So, A ∩ B ∩ C = {4, 6}

Question 1:

If A = {1,3,5,7,9}, B = {2,4,6,8} and C = {2,3,5,7,11}, find (A ∩ B) and (A ∩ C) What is your conclusion?

Solution:

Given that

A = {1, 3, 5, 7, 9}

B = {2, 4, 6, 8}

C = {2, 3, 5, 7, 11}

A ∩ B = {1, 3, 5, 7, 9} ∩ {2, 4, 6, 8}

= Φ

Therefore, A and B are disjoint sets

A ∩ C = {1, 3, 5, 7, 9} ∩ {2, 3, 5, 7, 11}

= {3, 5, 7}

Therefore, A and B are disjoint sets whereas A and C are intersecting sets.

Intersection of Two Sets

The intersection of two sets is the set of all the elements of two sets that are common in both of them. Intersection is related to grouping up the common elements.

If we have two sets A and B, then the intersection of them is denoted by A ∩ B and it is read as A intersection B.

Let X = {2, 3, 8, 9} and Y = {5, 12, 9, 16} are two sets.

Now, we are going to understand the concept of Intersection of set. It is represented by the symbol " ∩ ". Two sets A and B are said to be intersecting if A ∩ B $\neq$ Φ different sets.

If we want to find the intersection of A and B, the common part of the sets A and B is the intersection of A and B. It is represented as A ∩ B. That is, if an element is present in both A and B, then that will be there in the intersection of A and B. You will understand it better with the figure below.

If A and B are two sets. Then, the intersection of A and B can be shown as below.

The intersection of A and B can also be denoted by A ∩ B.

Therefore, A ∩ B = {x : x ε A and x ε B}.

Noticeably, x ε A ∩ B i.e., x ε A and x ε B

In the diagram above, the shaded area stands for A intersection B which is represented as A ∩ B.

In a similar manner, if A1, A2... An is a finite family of sets, then their intersection is represented by A1∩ A2 ∩......∩ An.

More on Intersection Of Two Sets

The intersection of two sets X and Y is the set of elements that are common to both set X and set Y. It is denoted by X ∩ Y and is read ‘X intersection Y’.

Example:

Draw a Venn diagram to represent the relationship between the sets

X = {1, 2, 5, 6, 7, 9, 10} and Y = {1, 3, 4, 5, 6, 8, 10}

Solution:

X ∩ Y = {1, 5, 6, 10} ← in both X and Y

For the Venn diagram,

Step 1 : We draw two overlapping circles to represent the two sets.

Step 2 : Next, we write down the elements in the intersection.

Step 3 : Write down the remaining elements in the various sets.

Observe that you start filling the Venn diagram from the elements in the intersection first.

If X ∩ Y then X ∩ Y = X. We will demonstrate this relationship in the following example.

Example:

Draw a Venn diagram to represent the relationship between the sets

X = {1, 6, 9} and Y = {1, 3, 5, 6, 8, 9}

Solution:

We discover that X ∩ Y = {1, 6, 9} which is equal to the set X

For the Venn diagram,

Step 1 : We draw one circle within another circle

Step 2 : We write down the elements in the inner circle.

Step 3 : We write down the remaining elements in the outer circle.

Intersection of Convex Sets

In a Vector space, a set is known as convex set if all the elements of the line connecting two points of that set as well lie on that set. In other words, we can say that the set S is convex set if for any points x, y ε S, there are no points on the straight line joining points x and y that are not in the set S.

The intersection of two convex set is once more a convex set. We can prove it with the aid of contradiction method. Therefore, let’s suppose that A and B are the two convex sets. And, let we have two points x and y in such a way that x ∈ A ∩ B and y ε A ∩ B, then x ε A, x ε B, y ε A and y ε B and there will be a point z in a manner that z is not in A or B or both. This is the contradiction of our assumption that A and B are the convex sets. So there is no such point x, y and z can exists and A ∩ B is a convex set.

Intersection of Three Sets

If we have A, B and C, then the intersection of these three sets are the set of all elements A, B and C that are common in these three sets.

Questions and solutions

Question:

If we have A = {1, 3, 5, 7, 6, 8}, B = {2, 4, 6, 8, 9} and C = {1, 3, 6, 8}, then find the A ∩ B ∩ C.

Solution:

Given that A = {1, 3, 5, 7, 6, 8}, B = {2, 4, 6, 8, 9} and C = {1, 3, 6, 8}.

Then, it is obvious that the elements 6 and 8 are common in all the three given sets.

Hence, we say A ∩ B ∩ C = {6, 8}.

More on Intersection Of Three Sets

Venn Diagrams of three sets

The intersection of three sets X, Y and Z is the set of elements that are common to sets X, Y and Z. It is denoted by X ∩ Y ∩ Z

Example:

Draw a Venn diagram to represent the relationship between the sets

X = {1, 2, 5, 6, 7, 9}, Y = {1, 3, 4, 5, 6, 8} and

Z = {3, 5, 6, 7, 8, 10}

Solution:

We find that X ∩ Y ∩ Z = {5, 6}, X ∩ Y = {1, 5, 6},

Y ∩ Z = {3, 5, 6, 8} and X ∩ Z = {5, 6, 7}

For the Venn diagram:

Step 1 : Draw three overlapping circles to represent the three sets.

Step 2 : Write down the elements in the intersection X ∩ Y ∩ Z

Step 3 : Write down the remaining elements in the intersections:

X ∩ Y, Y ∩ Z and X ∩ Z

Step 4 : Write down the remaining elements in the respective sets.

Again, observe that you start filling the Venn diagram from the elements in the intersection first.

Generally, there are various ways that 3 sets may intersect. A few examples are illustrated below.

Intersection of Open Sets

Every intersection of open sets is once more an open set. Let us have two open sets A1 and A2. If the intersection of both of them is empty and empty set is again an open set. Therefore, the intersection is an open set.

If A1 and A2 are open sets, then there will be a few x ε A1 ∩ A2. Since the given sets are open, we have a few r1 and r2 in a manner that $B_{r_{1}}(x)\subset A_{1}$ and $B_{r_{2}}(x)\subset A_{2}$. Thus, we can select a number $B_{r}(x)\subset A_{1}\cap A_{2}$.

Therefore, we can say that if the intersection is not empty, then by the use of definition of intersection and non emptiness, there will be x ε Ai present for all Ai's, where all Ai's are open sets. Subsequently, we have $B_{r_{i}}(x)\subset A_{i}$ for some ri > 0.

Complement Of The Intersection Of Sets and Symmetric Difference

In this section, we will discuss about the complement of the intersection of sets, the symmetric difference of two sets and the symmetric difference of three sets.

The complement of the set X ∩ Y is the set of elements that are members of the universal set U but not members of X ∩ Y. It is denoted by (X ∩ Y) ’.

The symmetric difference of two sets is the group of elements which are members of either set but not both - in other words, the union of the sets with the exception of their intersection. Forming the symmetric difference of two sets is simple, but forming the symmetric difference of three sets is a bit more difficult.

Example:

Suppose U = set of positive integers less than 10,

X = {1, 2, 5, 6, 7} and Y = {1, 3, 4, 5, 6, 8}.

a) Draw a Venn diagram to illustrate (X ∩ Y) ’

b) Find (X ∩ Y) ’

Answer:

a) First, we fill in the elements for X ∩ Y = {1, 5, 6}

Secondly, we fill in the other elements for X and Y and for U

Shade the region outside X ∩ Y to indicate (X ∩ Y) ’

b) We can observe from the Venn diagram that

(X ∩ Y) ’ = {2, 3, 4, 7, 8, 9}

Or we deduce that X ∩ Y = {1, 5, 6} and so

(X ∩ Y ) ’ = {2, 3, 4, 7, 8, 9} View more on psalmfresh.blogspot.com for more educating and tricks article Thank you