 Hello and welcome to this session. In this session we will discuss the following question and the question says, let A is equal to letters of the word mathematics and B is equal to letters of the word history. Find A union B, A intersection B and represent them by Venn diagrams. Let's start the solution now. We are given A is equal to letters of the word mathematics. Let us first list down the elements of the set A. Since the repetition of the elements is not allowed in the dilatation of a set, so the set A is equal to the set containing the letters m, a, t, h, e, i, c, s. That is the set A in roster form is the set containing the letters m, a, t, h, e, i, c, s. Also we are given in the question that B is equal to letters of the word history. So the set B in roster form is the set containing the letters h, i, s, t, o, r, y. We can see that the common elements between the sets A and B are t, h, i, s. First we will find out what is A union B. We know that the set A union B contains elements which are in the set A or in B or both. So A union B contains the elements of the set A and B and the common elements are written only once. So A union B is equal to the set containing the elements m, a, t, h, e, i, c, s, o, r, y. We shall now draw the well diagram for A union B. First draw two intersecting circles which represent the sets A and B. We have drawn intersecting circles because the sets A and B have common elements between them. So they are overlapping sets. We now write down the common elements between the sets A and B in this common portion of these overlapping circles. So we write the elements i, t, h, s in this common portion. Next we write down the remaining elements of the set A in this remaining portion of the circle A, the remaining elements of the set A are m, a, e, c. So we write down these elements in this remaining portion of the circle A. Next we write down the remaining elements of the set B in this remaining portion of the circle B. The remaining elements are o, r, y. So we write down these elements in this remaining portion of the circle B. So in this way we have written down the elements of the sets A and B in the two circles A and B. Now since A union B contains elements which are in A or in B or both, so this shaded portion is A union B. Let us now write down what we have represented. Union of the two sets A and B that is A union B is represented by the shaded portion. We will now find what is A intersection B. Now A intersection B contains elements which are in both the sets A and B that is the set A intersection B contains common elements between the two sets A and B. So A intersection B is equal to the set containing the elements i, t, h, s. We now draw the Venn diagram for A intersection B. As done earlier we draw two intersecting circles representing the sets A and B. Then we write down the common elements of the sets A and B in this common portion between the two circles and then write down the remaining elements of the set A and the set B in this non-common portion of the circle A and circle B respectively. This common portion between the two circles A and B contains elements which are in set A and in set B. So we shape this common portion. This shaded portion is A intersection B. We now write down what we have represented. The set A intersection B that is intersection of the two sets A and B is represented by the shaded portion. So we have found what is A union B and A intersection B and represented them by Venn diagrams. With this we end our session. Hope you enjoyed the session.