Question

if P={x:x€W and 0<x<10} Q={x:x=2n+1n€W and n<5} and R={2,3,5,7,11,13} than verify P-(QnR)=(P-Q)U(P-R)​

Answer 1

Answer:

Given:

P= {x : x€W & 0<x<10}.

Q= {x: x=2n+1n , n€W & n<5}.

R= {2 , 3 , 5 , 7 , 11 , 13}

[Rough column:

Q=2×0+1×0=0

Q=2×1+1×1 = 3

Q=2×2+1×2=6

Q=2×3+1×3=9

Q=2×4+1×4=12]

P= {1 ,2 ,3 ,4 ,5 , 6, 7, 8, 9}

Q={0 , 3 , 6 , 9 , 12}

R={2 , 3 , 5 , 7 , 8 , 9}

LHS: P-(QnR)

let us find (QnR) first

(QnR) = {0 ,3 ,6 ,9 ,12} n {2, 3, 5, 7, 11, 13}

(QnR) = {3}

let us find P-(QnR) Now

P-(QnR)={1, 2, 3, 4, 5, 6, 7, 8, 9,} - {3}

P-(QnR)={1, 2, 4, 5, 6, 7, 8, 9} ----- ( 1 )

RHS:(P-Q)U(P-R)

let us find (P-Q) first

(P-Q) = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {0, 3, 6, 9, 12}

(P-Q) = {1, 2, 4, 5, 7, 8}

let us find (P-R) now

(P-R) = {1, 2, 4, 5, 7, 8} - {2, 3, 5, 7, 11, 13}

(P-R) = {1, 4, 6, 8, 9}

Now let's find (P-Q)U(P-R)

(P-Q)U(P-R) = {1, 2, 4, 5, 7, 8}U{1, 4, 6, 8, 9}

(P-Q)U(P-R) = {1, 2, 4, 5, 6, 7, 8, 9} ------( 2 )

From ( 1 ) & ( 2 ) we can see that LHS=RHS

P-(QnR) = (P-Q)U(P-R)

Hence We Verified