 Hello and welcome to the session on Algebraic Structures under the course Discrete Mathematical Structures at 2nd year of Information Technology Engineering semester 1. At the end of this session students will be able to demonstrate algebraic systems and its properties. These are the contents, algebraic systems, definition and examples and some simple algebraic systems and general properties. To start with, let us see how an algebraic system is defined. A system consisting of a set and one or more n array operations on the set given by S, F1, F2 etc. where S is a non-empty set and F1, F2 etc. are operations on S. Two algebraic systems capital X comma dot and capital Y comma star are said to be of the same type whenever the n array operations dot and star have the same value of n. The algebraic system capital I where I denotes the set of integers comma plus comma into where plus and into are the operations of addition and multiplication respectively on I. Capital A denotes operation of addition and let capital M denotes operation of multiplication. Now let us define certain properties that this algebraic system is going to have. Those are numbered from A1, A2 etc. where A stands for addition. Let us see the properties of addition. A1 is a property saying for any A, B, C which belongs to I we get A plus B plus C is equal to A plus B plus C and we all know that this is a property known as associativity. A2 is a property for any A comma B belongs to I, A plus B is equal to B plus A and it is known as commutativity. A3 is a property thereby there exist a distinguished element 0 belongs to I such that for any A belongs to I A plus 0 is equal to 0 plus A is equal to A and it is called as an identity element. Such A4 for each A belongs to I there exist an element in I denoted by minus A and called negative of A such that A plus minus A is equal to 0 and it is known as the inverse element. Similarly, we have the properties of multiplication given by M1 as for any A, B, C belongs to I we have A into B into C is equal to A into B into C again it is known as associativity M2 for any A comma B belongs to I A into B is equal to B into A known as commutativity. Next M3 there exist a distinguished element 1 belongs to I such that for any A belongs to I A multiplied by 1 is equal to 1 multiplied by A is equal to A and it is called as an identity element. Next capital D for any A, B, C belongs to I A into in bracket B plus C is equal to A into B plus A into C and we know that this property is called as distributivity. The operation multiplication distributes over the operation of addition and the last capital C for any A, B, C belongs to I and A not equal to 0 we get A multiplied by B is equal to A multiplied by C and it implies B must be same as C and we call this as the cancellation property. Here is the next example for you the algebraic system capital R comma plus comma into where again plus and into are the operations of addition and multiplication on R and capital R is the set of real numbers. Now pause the video for a while and try to answer this question what are the properties of algebraic system R comma plus comma into system R comma plus comma into also satisfies all the properties of the system I comma plus comma into. Now as you have seen number of examples which explains what is an algebraic system along with its properties try to solve this assignment consider the set capital B is equal to set of 0 comma 1 and the operations plus and into on B given by the following tables. The tables denote if you perform the operation of addition with these two elements as 0 and 1 what are the results similarly the second table demonstrates what may be the results after performing the multiplication of these two numbers. Second or the first table for addition we get 0 plus 0 and the answer is 0 0 plus 1 equal to 1 1 plus 0 equal to 1 and 1 plus 1 is equal to 0 similarly from the table of multiplication we get 0 into 0 equal to 0 0 into 1 equal to 0 1 into 0 also equal to 0 whereas 1 multiplied by 1 the result is 1. Now for this algebraic system show that B comma plus comma into satisfies all the properties of I comma plus comma into where I is a set of integers and we have also seen the various properties it holds for addition and multiplication. Apply the same theory and try to list out all the properties for this algebraic system as that of the set of integers. Moving on to some more definitions which are based upon the algebraic system. Let X comma dot and Y comma star be two algebraic systems of the same type. Then a mapping G given by X to Y is called a homomorphism or simply morphism from X comma dot to Y comma star if for any X1 comma X2 which belongs to the first set capital X we have G of X1 dot X2 is equal to G of X1 multiplied by or rather star G of X2. Now you can notice here the left hand side denotes the dot operation and the right hand side uses the other operation as star and Y comma star is called a homomorphic image of X comma dot. Next here the definition let G be a homomorphism the definition of which we have seen in the earlier slide from X comma dot to Y comma star. Now if G which is a mapping from X to Y is on to then G is called a monomorphism. In the earlier slides we have seen the types of mappings or the functions being one to one on to one to one on to and so on. So, out of which if you find that the mapping G is on to then it is known as monomorphism. So, monomorphism is nothing but an extension of homomorphism being G as on to if G X to Y is one to one on to then we name it separately as isomorphism and X comma dot and Y comma star are said to be isomorphic. Next let X comma dot and Y comma star be two algebraic systems such that Y is a subset of X. Now a homomorphism G from X comma dot to Y comma star in such a case is called an endomorphism and if Y is equal to X then an isomorphism from X comma dot to Y comma star is called an automorphism. So, there is a slight difference when you lead from homomorphism to endomorphism when Y is a subset of X and automorphism when X and Y are seen. If G is one to one on to then G is called an isomorphism and X comma dot and Y comma star are said to be isomorphic. These are the references I hope with this you must have understood what is an algebraic system and also the related concepts given by the various definitions on algebraic systems. Thank you.