 Hello and welcome to the session. In this session we will learn about Cartesian product of more than 2 sets, inverse of a relation and value of a function. First of all let us learn about Cartesian product of more than 2 sets. Now if there are 3 sets a, b and c then using 3 elements b and c such that a belongs to the set a, b belongs to the set b and c belongs to the set c we form an ordered triplet a, b, c. Now the set of all such ordered triplets is called the Cartesian product of the 3 sets a, b and is denoted by a cross b cross c. Now let us discuss one example here let a is a set containing the elements 1, 2 and 3, b is a set containing the elements 4, 5, 6 and 7 and c is the set containing the elements 8 and 9 and a cross b cross c. Now using this definition which we have discussed earlier a cross b cross c will be equal to the sets containing the ordered triplets 1, 4, 8 that is 1 is the element of the set a, 4 is the element of the set b and 8 is the element of set c. Now here we can form the other ordered triplets which are 1, 4, 9, 1, 5, 8, 1, 5, 9, 1, 6, 8 that is choosing the first component of the ordered triplet from the set a, second component from the set b and third component from the set c. So we have formed all the ordered triplets where 1, 2 and 3 belong to the set a, 4, 5, 6 and 7 belong to the set b and they belong to the set c. Similarly the Cartesian product of n sets a1, a2 and so on up to an is denoted by a1 cross a2 cross cross an consists ordered n triples a1, a2 and so on up to an where a1 belongs to the set a1, a2 belongs to the set a2 and so on up to an belongs to the set an. Symbolically we can write that the product over ai where i varies from 1 to n is equal to the set of all ordered n triples that is a1, a2, so on up to an such that ai belongs to the set ai and i varies from 1 to n. Also the number of elements in a1 cross a2 cross so on up to cross an is equal to number of elements in a1 into number of elements in a2 so on up to into number of elements in an where i is finite for 1 is less than equal to i less than equal to n that is where i varies from 1 to n. For example in this case set a is having set b is having 4 elements and set c is having 2 elements so number of elements into 2 which is equal to 24 that is number of elements in a cross b cross c is equal to number of elements in a into number of elements in b into number of elements in c which is equal to 24. Now here you can also check that there are 24 ordered triplets in this set that means there are 24 elements in this set that is a cross b cross c therefore number of elements in a cross b cross c is equal to number of elements in a into number of elements in b into number of elements in c. Now let us discuss inverse of a relation. Now for any binary relation the relation can be constructed by many interchangeable in every ordered pair in the given relation and the relation that is obtained is called the inverse of the given relation. It is denoted by that is the inverse of r is denoted by r inverse and symbolically r inverse is equal to the set containing the ordered pair yx where the ordered pair xy belongs to the set that is to the relation r. Now let us discuss one example now let r be a set containing the ordered pairs ax by now we can find r inverse by interchanging the first and the second components in every ordered pair in the given relation. So this is equal to the set containing the ordered pairs xa, yb. Now let us discuss value of a function. Now function is a rule by which we associate the element of one set to the element of the other set. Now if f is a function a to b then y is equal to f of x. Now we use f of x to denote the object which f associates with x that is f of x is the functional value to also designate the function f from a to b into f of x as one example for this. Here let f be a function that is it is a set containing the ordered pairs ab, cd and domain of the first components of all the ordered pairs in the given function. So it will be the set containing the elements a, c and b and the range of the function will be equal to the set containing the second components of all the ordered pairs in the given function. Let us it is the set containing the elements v, d and k. So set with the help of the arrow diagram. Now here we have drawn two circles in the first circle we will write the domain of the function that is a, c and b and in the second circle we will write the range of the function that is b, d and q. Now a is related with b, c is related with d and p is related with q. Now this is a function from the first set to the second set. Now from the arrow diagram you can see that b is the value of the function at a therefore b is equal to f of a. Now d is the value of the function at c therefore d is equal to f of c and here q of the function at p therefore q is equal to f of, so in this way we have discussed the value of the function with the help of arrow diagram. So in this session we have learnt about partition product of more than two sets, inverse of a relation and value of a function. So this completes our session. Hope you all have enjoyed the session.