18. P is the solution set of 7x - 2 > 4x + 1 and
Q is the solution set of 9x - 45 > 5 (x - 5);
where x e R. Represent :
(i) PnQ
(ii) P - Q
(ii) Pn Q' on different number lines.
Answers
Step-by-step explanation:
Given :-
P is the solution set of 7x - 2 > 4x + 1
and Q is the solution set of 9x - 45 > 5 (x - 5);
where x e R.
To find :-
Represent :
(i) PnQ
(ii) P - Q
(ii) Pn Q' on different number lines.
Solution :-
Given that
P is the solution set of 7x - 2 > 4x + 1
=> 7x - 2 > 4x + 1
On adding 2 both sides
=> 7x -2 + 2 > 4x+1 +2
=> 7x > 4x+3
On subtracting 4x both sides then
=> 7x-4x > 4x-4x+3
=> 3x > 0+3
=> 3x > 3
On Dividing by 3 both sides
=> (3x/3) > (3/3)
=> x > 1
P = { 2,3,4,...}, x€ R --------(1)
and
Q is the solution set of 9x - 45 > 5 (x - 5)
=> 9x - 45 > 5 (x - 5)
=> 9x -45 > 5x -25
On adding 45 both sides then
=> 9x -45+45 > 5x -25+45
=> 9x +0 > 5x+20
=> 9x > 5x+20
On Subtracting 5x both sides then
=> 9x -5x> 5x-5x+20
=> 4x > 0+20
=> 4x > 20
On dividing by 4 both sides then
=> (4x/4) > (20/4)
=> x > 5
Q = { 6,7,8,...} ,x€ R --------(2)
Now,
I)PnQ :-
=> { 2,3,4,...} n { 6,7,8}
=> { 6,7,8...}
PnQ = { 6,7,8...} = Q
ii) P-Q :-
=> { 2,3,4,...} - { 6,7,8...}
=> { 2,3,4}
P-Q = { 2,3,4}
iii) PnQ':-
Q' = U - Q
Where U is the universal set that is set of Real numbers
Q' = { ...,-2,-1,0,1,2...} - { 6,7,8...}
=> Q' = { ...,-2,-1,0,1,2,3,4,5}
Now ,
PnQ' = {2,3,4,..} n { ...,-2,-1,0,1,2,3,4,5}
=> {2,3,4}
PnQ' = {2,3,4}
Answer :-
I)PnQ = { 6,7,8...} = Q
ii) P-Q = { 2,3,4}
iii)PnQ' = { 2,3,4}
Used formulae:-
- Let A and B are two non empty sets then The set of Common elements in both the sets A and B is called Intersection of A and B .It is denoted by AnB.
- Let A and B are two non empty sets then The set of elements which are belongs to only A is called the difference of the two sets A and B .It is denoted by A-B.
- All the number sets are subsets of Real numbers, So Real numbers is the Universal set and the universal set is denoted by U.
- A is the set and U is the universal set A' = U-A.
