 Hello and welcome to the session. In this session we will describe events as a subset of sample space using characteristics of the outcomes like union, intersection and complement of an event. We will make use of main diagrams to understand union, intersection and complement of events. In our earlier sessions we have learned about the meaning of sample space and event of an experiment. We know that event is always a subset of the sample space in probability. The universal set is the sample space of the experiment and circle represents particular event. Now suppose we have a sample space equal to a set containing elements 1, 2, 3, 4, 5 and 6 and yet outcomes of an event E is a set containing elements 1 and 2. So we can show it in a main diagram. For this we draw a rectangular box which represents sample space S and we write the elements of sample space in this rectangular box. Now event E is the subset of sample space elements of event E are 1 and 2. So we draw a circle inside the rectangle which includes only elements 1 and 2. So here we have drawn a circle which represents event E we will write elements 1 and 2 inside this circle. Now all the other elements of sample space S which are not in event E lie outside this circle. So we write here 3, 4, 5 and 6 and we share the circle representing event E which represents sample space S and event E which is a subset of sample space S. Now let us discuss union of events. Now let A and B be any two events. Now union of these two events is the set of elements which are in both A and B and here the repeated elements are written only once and it is denoted by union B. Now we show it with the help of main diagram. Now let sample space S is a set containing elements 1, 2, 3, 4, 5, 7 comes for event E is a set containing elements 1, 5, 2, 3 and event E is a set containing elements 1, 4, 5 and 7. Then in union B is a set of elements which are in A or in B or in both A and B and here the repeated elements are written only once. So this is equal to set containing elements 1, 2, 3, 4, 5 and 7. Now here you can see the repeated elements 1 and 5 are written only once. Now all the elements of set A and union B are contained in sample space S event A union B is a subset of sample space. Now let us draw its main diagram. Here first we have drawn a rectangle which represents sample space we will draw two circles representing set A and set B and these two circles will be intersecting circles because set A and set B have elements 1 and 5 in common. So here we have drawn two intersecting circles representing set A and set B. Now in this intersecting portion we will write the common elements that is we will write 1 and 5 here. Then we will write the remaining elements of set A and set B. So we will write 2 and 3 here and 4 and 7 here. Now all the other interface S which are neither in set A nor in set B lie outside the two circles. So here the only element is 8. So we will write 8 outside the two circles. Now here we have to represent A union B. Now here the shaded portion represents A union B. Also you must note that A union B means either or for example we can be asked find the probability that either of the events occur or find probability P of the event A or B. Now let us discuss intersection of events. Now let A and B be any two events then intersection of these two events is the set of elements which are common to both A and B that is we include only those outcomes which are in event A as well as in event B which is denoted by intersection B like in this example which we have discussed earlier we see that the elements 1 and 5 are in both the events A and B thus 1 and 5 are common outcomes to both the events thus A intersection B is a set between the elements 1 and 5. Also we see that A intersection B is a subset of central space S. Now let us draw its ring diagram here the diagram is same as above but the shaded portion will be different and we shade only that part the circles which is common to both A and B. So here this is the required ring diagram where the shaded portion represents A intersection B. Now you must know that in questions it can be asked find the probability of getting both A and B events of A and B that is probability of event A and B it means here we have to find probability of the event A intersection B. Now let us discuss complement of an event. Now let E be any event. Now complement of an event E is the set of those elements of central space S which are not in event E and it is denoted by E complement. Now earlier we have discussed an example where central space S is a set containing elements 1, 2, 3, 4, 5 and 6 and event E is a set containing elements 1 and 2. Now here E complement will be a set containing those elements of central space S which are not in event E and those elements are 3, 4, 5 and 6. So E complement is a set containing elements 3, 4, 5 and 6. The elements 3, 4, 5 and 6 are not in event E. Now earlier we have drawn a ring diagram representing central space S and event E which is a subset of central space S. Now here we have to represent E complement. Now here the shaded portion in this ring diagram represents E complement. Now you must note that we can be asked to find probability of an event not E or to find probability of an event E complement. In both these cases we have to find probability of the event E complement. Now let us discuss complement of two events. Now complement of set A union B is the set of those elements of central space S which are not in set A union B and it is denoted by A union B whole complement. Now from this ring diagram can we find A union B whole complement. Now only element 8 is outside the two circles representing A union B. This means 8 is not in complement is a set containing. Now let us represent A union B whole complement in this ring diagram. So we have the following ring diagram and here shaded portion represents B whole complement. Now you must note that if it is asked to find probability of an event neither in A nor in B then we have to find probability P of an event A union B whole complement equivalently it also means not in A and not in B. So we can find of an event A complement and B complement or probability P of an event A complement intersection B complement. So in this session we have discussed union intersection and complement of events and this completes our session. Hope you all have enjoyed the session.