 Hello and welcome to the session. In this session, we will understand the concept of relations and functions. And we will also see what is a domain and range. Now let us see the concept of relation. Now relation has a set of ordered pairs. For example, we have two sets, set P, which is a set containing the elements 1, 2 and 3 and set Q, which is a set containing the elements A, B and C. Now we will write a set containing all the ordered pairs formed from the set P and Q such that first element is taken from set P and second element from set Q. So we obtain the following set of ordered pairs. Here you can see that in all the ordered pairs, first element is taken from set P and second element from set Q. So set S is a relation from set P to set Q. Now let us take any subset of set S, let it be C, which is a set containing the ordered pairs 1, B, 1, C. So as it is a subset, that is set C is also a relation from set P to set Q. Thus a relation from set P to set Q is a set of ordered pairs x, y, where x belongs to set P and y belongs to set Q. Now let us see what is a domain of a relation. Now domain of a relation is a set of possible values that x may have, that is set of h and now let us see what is range of relation. Now range of relation is a set of possible values that y may have, that is set of y coordinates and in writing domain and range we do not repeat values. Now for this relation let us find domain and range. Now here domain is equal to possible values that x may have, we can say domain is a set containing the first components are 1, 1 and 2. Now we do not repeat the values, so domain is a set containing the elements 1 and range is a set of possible values that y may have. Or we can say range is a set containing the second components range is equal to set containing the elements b, c and domain is the set of bad values which x coordinate takes and range is the set of those values which y coordinate takes in the ordered pair. Now we can draw a relation by plotting the ordered pairs on a coordinate plane. We can show the relation in the form of a table. Now let us discuss mapping. Now we can show a relation using mapping. Here we will draw two circles, list the values of x that is domain in the first circle, the left circle use of y that is range in the right circle and elements should not be repeated. Then draw arrows from each value of x to the corresponding value of y. Now consider this relation from set a to b. Now domain of the above relation is the set containing the elements minus 2, 1, 2, 0 and 3. That is domain is a set containing all the ordered pairs and range is a set containing second components of all the ordered pairs. So range is a set containing the elements 6, 4, 0, y is 6. Now let us name the first circle as a and second circle as b. Now let us list the values of x that is domain in set a vertically. So it will be minus 2, 1, 2, 0, 3 and let us list the values of y that is range in set b. So it will be 6, 4, 0, minus 6. Now minus 2 is related with 6 so we will draw an arrow from minus 2 from 1 to we have drawn the remaining arrows and we get the required mapping. Now let us discuss what is a function. Now a function is a relation in which members of the domain do not repeat. Every value of x there is only one corresponding value of y. Here y values can be repeated but x values cannot be repeated. Now consider this relation. Now let us check whether this relation is a function or not. Now here in these two ordered pairs you can see that these two ordered pairs have same x values. These two ordered pairs have same x values to ordered pairs. Same first members and one are repeated. This relation is not a function. Now consider another relation. Now here you can see the x values are different so here no x value is repeated therefore this relation is a function. Now let us see the definition of a function. Now a function is a relation in which each element of the domain is paired with exactly one element of range. Now we can identify a function from table or mapping or set of ordered pairs of the given relation. Now let us see how to identify functions using vertical line test. Now we can use a vertical line test to identify whether the given graph represents a function or not. Now in vertical line test if a vertical line is passed over the graph and it intersects the given graph in exactly one point then given graph is a function. And if the vertical line intersects the graph at more than one point then it is not a function. Now consider the following graphs. Now let us draw vertical line on both graphs. Now here we have drawn a vertical line in both graphs and here you can see that in first graph line intersects the given graph at only one point so it is a function but in second graph the line intersects the given graph at two points it is not a function. Now let us discuss domain and range of a function. Now we know that domain is a set of possible values that x may have and range is a set of possible values that y may have. Now we write the domain and range of a function in intervals. For example consider the line segment from the point with coordinates 1, 3 to the point with coordinates 5 minus 2. Now joining these points we get a graph of straight line. Now if we draw a vertical line it will intersect the given graph that is this straight line at only one point so it is a function. We started the line from x is equal to 1 till x is equal to 5. So x can take any value between x is equal to 1 and x is equal to 5 that is x varies from 1 to 5 it means x can take any real value between this interval so domain is the set of all x such that 1 is less than equal to x is less than equal to 5. y varies from minus 2 so here minus 2 is less than equal to y is less than equal to 3. That means y can take any real value between this interval so here range is the set of all y such that minus 2 is less than equal to y is less than equal to 3. So in this session we have discussed the concept of relations and functions and this completes our session. Hope you all have enjoyed the session.