 Hello and welcome to this session. In this session, first we will see the introduction of Boolean Algebra. Earlier it was observed that the Boolean Algebra was used in switching circuits and now the Boolean Algebra has found its application in the designing of computers which are based on the binary numbers 0 and 1. They correspond to off respectively in switching circuits. So we can say that Boolean Algebra is the two-valued algebra and this Boolean Algebra or the two-valued algebra was applied earlier to statements which were either true or false. It is applied to switches which are either closed or you can say on or open that is off. Next we discuss three basic operations in Boolean Algebra. First basic operation is and in case of the logical statements it is symbolized as conjunction and in case of theory of sets it is symbolized as intersection. Next basic operation is or in case of logical statements it is symbolized as disjunction and in case of theory of sets it is symbolized as union. Then we have the operation not in logical statements it is symbolized as negation and in theory of sets it is symbolized as the complement of the set. Further for the operation of and a dot would be used for the operation of or plus would be used and for the operation of not a prime would be used. Now let's discuss the laws of sets the principle of duality plays an important role in the algebra of sets the Boolean Algebra. Like in set theory union that is denoted by this symbol and intersection which is denoted by this symbol are the dual operations that is in a statement if we interchange the union and the intersection then also the new statement does obtain would be true and this complementation that is the symbol is a set doled and the universal set and the empty set are the dual elements. Let's discuss the primary laws in this first we have set A union set A is equal to set A then next we have set A intersection set A is equal to set A these two are the idempotent laws and by using the principle of duality we can get the second law from the first law then next we have A union empty set phi is equal to A then we have A in the section universal set psi is equal to A. Now this fourth law can be obtained from this third law using the principle of duality since we know that the universal set and the empty set are the dual elements and union and intersection are dual operations. Next law that we have is A union the universal set psi is equal to the universal set psi then next we have A intersection empty set phi is equal to the empty set phi. Here also the sixth law can be obtained from the fifth law using the principle of duality. Next we discuss the associative laws according to this associative law we have A union B the whole union C is equal to A union B union C the whole. Then next we have A intersection B the whole intersection C is equal to A intersection B intersection C the whole. Now here also the second law can be obtained from the first law using the principle of duality as union and intersection are the dual operations. Next let's discuss the commutative laws first we have A union B is equal to B union A then we have A intersection B is equal to B intersection A. Here also the second law can be obtained from the first law using the principle of duality. Next we discuss the distributive laws this we have A union B intersection C the whole is equal to A union B the whole intersection A union C the whole. Next we have A intersection B union C the whole is equal to A intersection B the whole union A intersection C the whole. In this case also the second law can be obtained from the first law using the principle of duality. Now next we have the laws of complements in this the first law is A union A complement is equal to the universal set psi then next we have A intersection A complement is equal to the empty set phi. Here the second law is obtained from the first law using the principle of duality. Next law is phi complement is equal to the universal set psi then we have the universal set psi complement is equal to phi. In this case also the fourth law can be obtained from the third law using the principle of duality. Next law that we have is A union B the whole complement is equal to A complement intersection B complement. Next we have A intersection B the whole complement is equal to A complement union B complement. Now these two laws are the de Morgan's laws and here the sixth law is obtained from the fifth law using the principle of duality. Next law that we discuss is A complement the whole complement is equal to A as we know that complementation is a self dual. So this law is obtained from itself using the principle of duality then we have A union A intersection B the whole is equal to A. Next is A intersection A union B the whole is equal to A. Here this minus law is obtained from the eighth law using the principle of duality. Next we discuss the comparison of laws of sets and arithmetic. First let's see the comparison of the commutative law in arithmetic and in laws of sets. In arithmetic out of the four binary operations only addition and multiplication are commutative like if we have two numbers x and y then x plus y is equal to y plus x and x into y is equal to y into x. This is the commutative law in arithmetic which shows that addition and multiplication are commutative. Subtraction and division do not obey the commutative law and according to the laws of sets the commutativity of addition and multiplication are resembled by the laws. A union B is equal to B union A and A intersection B is equal to B intersection A. Next we see the comparison of the associative law in arithmetic and laws of sets. In arithmetic again addition and multiplication are associative that is x plus y the whole plus z is equal to x plus y plus z the whole where x, y and z are three numbers then x into y the whole into z is equal to x into y into z the whole. Now according to the laws of sets we have two laws which resemble these associative laws of arithmetic. So here we have A union B the whole union C is equal to A union B union C the whole. Then we have A intersection B the whole intersection C is equal to A intersection B intersection C the whole. Next law is the distributive law we know that in arithmetic multiplication distributes over addition that is we have x into y plus z the whole is equal to x into y plus x into z. And in the law of sets we have a law which is similar to this law in arithmetic that is A intersection B union C the whole is equal to A intersection B the whole union A intersection C the whole. For an arithmetic addition does not distribute over multiplication that is x plus y into z the whole is not equal to x plus y the whole into x plus z the whole. But in the law of sets union distributes over intersection that is we have A union B intersection C the whole is equal to A union B the whole intersection A union B the whole. This completes the session hope you understood the introduction of Boolean algebra and the laws of sets.