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Fig 1.10
12 Practica
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(ii) An A' =0
(i) A UA' = U
1. Complement laws:
2. De Morgan's law:
3. Law of double complementation : (A')' = A
4. Laws of empty set and universal set ' = U and U' = 0.
These laws can be verified by using Venn diagrams.
Let A
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EXERCISE 1.5
In e
(vi) (B – C)
(Fig 1.
If U = { a, b, c, d, e, f, g, h}, find the complements of the following sets -
(i) A = {a, b, c}
(ii) B = {d, e, f, g}
(iii) C = {a, c, e, g} (iv) D = { f, g, h, a}
of the following sets:
The complement A of a set A can be represented
The shaded portion represents the complement of the set A.
Let U= { 1, 2, 3, 4, 5, 6, 7, 8, 9), A = { 1, 2, 3,4}, B = { 2, 4,6,8 ) and
C = { 3, 4, 5, 6 }. Find (i) A' (ii) B' (iii) (A UC)(iv) (A UB)' (v) (Ay Note th=
by a Venn diagram as shown in Fig 1.10.
Some Properties of Complement Sets
(i) (A UB)' = A' B' (ii) (A n B) = Also be used i
Taking the set of natural numbers as the universal set, write down the complemcomplement A=a, b, c
Answers
Answer:
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; and
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; and
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; and
not (A and B) = not A or not B
In set theory and Boolean algebra, these are written formally as
{\displaystyle {\begin{aligned}{\overline {A\cup B}}&={\overline {A}}\cap {\overline {B}},\\{\overline {A\cap B}}&={\overline {A}}\cup {\overline {B}},\end{aligned}}}{\displaystyle {\begin{aligned}{\overline {A\cup B}}&={\overline {A}}\cap {\overline {B}},\\{\overline {A\cap B}}&={\overline {A}}\cup {\overline {B}},\end{aligned}}}
where
A and B are sets,
A is the complement of A,
∩ is the intersection, and
∪ is the union.
In formal language, the rules are written as
{\displaystyle \neg (P\lor Q)\iff (\neg P)\land (\neg Q),}\neg (P\lor Q)\iff (\neg P)\land (\neg Q),
and
{\displaystyle \neg (P\land Q)\iff (\neg P)\lor (\neg Q)}\neg (P\land Q)\iff (\neg P)\lor (\neg Q)
where
P and Q are propositions,
{\displaystyle \neg }\neg is the negation logic operator (NOT),
{\displaystyle \land }\land is the conjunction logic operator (AND),
{\displaystyle \lor }\lor is the disjunction logic operator (OR),
{\displaystyle \iff }\iff is a metalogical symbol meaning "can be replaced in a logical proof with".
Applications of the rules include simplification of logical expressions in computer programs and digital circuit designs. De Morgan's laws are an example of a more general concept of mathematical duality.