|
|
|
|
|
|
|
|
|
|
Prove that any two expression is equal or not calculator
|
1. A = {x<=5; x in N}; B = {2<=x<=8; x in N}; C = {x<=3; x in N}
2. A = {x^2<10; x in N}; B = {x<=1; x in N}; C = {1<=x<5; x in N}
3. A = {1,2,3,4}; B = {3,4,5}; C = {1,3,5}
4. A = {2,3,6}; B = {3,5,6}; U = {1,2,3,4,5,6}
5. A = {x<=9; x in N}; B = {3<=x<=7; x is odd}; C = {1< x<7; x is even}
6. A = {|x^3-2|<=25;x in N}; B = {1< x<5; x in N}; C = {x^4=81;x in N}
|
Example1. A = {1,2,3,4}, B = {3,4,5}, C = {1,3,5}, Prove that A intersect (B union C) = (A intersect B) union (A intersect C) ...
Solution:
Here A={1,2,3,4},B={3,4,5},C={1,3,5}
To find LHS = A ∩ (B uu C)
B uu C = {3,4,5} uu {1,3,5}
= {color{red}{3},4,color{green}{5}} uu {1,color{red}{3},color{green}{5}}
= {3,4,5,1}
A nn (B uu C) = {1,2,3,4} nn {3,4,5,1}
= {color{red}{1},2,color{green}{3},color{blue}{4}} nn {color{green}{3},color{blue}{4},5,color{red}{1}}
= {1,3,4}
:. A ∩ (B uu C) = {1,3,4} ->(1)
To find RHS = (A ∩ B) uu (A ∩ C)
A nn B = {1,2,3,4} nn {3,4,5}
= {1,2,color{red}{3},color{green}{4}} nn {color{red}{3},color{green}{4},5}
= {3,4}
A nn C = {1,2,3,4} nn {1,3,5}
= {color{red}{1},2,color{green}{3},4} nn {color{red}{1},color{green}{3},5}
= {1,3}
(A ∩ B) uu (A ∩ C) = {3,4} uu {1,3}
= {color{red}{3},4} uu {1,color{red}{3}}
= {3,4,1}
:. (A ∩ B) uu (A ∩ C) = {3,4,1} ->(2)
From (1) and (2)
:. A ∩ (B uu C) = (A ∩ B) uu (A ∩ C) (proved)
|
|
|
|
|
|
