Question

PROVE THE FOLLOWING


COMMUTATIVE LAW: A+B=B+A

A.B=B.A


DE MORGAN’S LAW: (A+B)’=A’.B’

(A.B)’=A’+B’​

Answer 1

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Commutative law ,

in mathematics, either of two laws relating to number operations of addition and multiplication, stated symbolically: a + b = b + a and ab = ba. From these laws it follows that any finite sum or product is unaltered by reordering its terms or factors. While commutativity holds for many systems, such as the real or complex numbers, there are other systems, such as the system of n × n matrices or the system of quaternions, in which commutativity of multiplication is invalid. Scalar multiplication of two vectors (to give the so-called dot product) is commutative (i.e., a·b = b·a), but vector multiplication (to give the cross product) is not (i.e., a × b = −b × a). The commutative law does not necessarily hold for multiplication of conditionally convergent series

De Morgan's laws

In propositional logic and Boolean algebra, De Morgan's laws[1][2][3] are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British mathematician. The rules allow the expression of conjunctions and disjunctions purely in terms of each other via negation.The rules can be expressed in English as:

the negation of a disjunction is the conjunction of the negations

the negation of a conjunction is the disjunction of the negations

or

the complement of the union of two sets is the same as the intersection of their complements

the complement of the intersection of two sets is the same as the union of their complements

or

not (A or B) = (not A) and (not B)

not (A and B) = (not A) or (not B),

where "A or B" is an "inclusive or" meaning at least one of A or B rather than an "exclusive or" that means exactly one of A or B.

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