Question

A={x:x positive integers and x²<18}
P={3,5,7}
Q={5,7}
and R=P/Q

1)Express A in roster method.
2)Find→(A×R)∩(A×Q)
3) Find P(A).If the element number of A is n then show that,the elements of P(A) support 2^n.

Answer with explanation.Don't spam please.​

Answer 1

$\boxed{1}$Positive integers are 1,2,3,4,5, . . . . . . . .

Here,

if x = 1, x² = 1² = 1

if x= 2, x² = 2² = 4

if x = 3, x² = 3² = 9

if x = 4, x² = 4² = 16

if x = 5, x² = 5² = 25 ; which is greater than 18

Accroding to question, the acceptable positive integers are 1,2,3 and 4.

$\begin{gathered}\;\;\;\;\underline{\boxed{\tt{Determinable\:set\:A={1,2,3,4}}}}\end{gathered}$

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$\boxed{2}$Given,

• P = {3,5,7}
• Q = {5,7}
• R = P/Q

R

= P/Q

= {3,5,7} - {5,7}

= {3}

We get from (1) that,

• A = {1,2,3,4}

To find : (A×R)∩(A×Q)

• Now,

(A×R)

= {1,2,3,4}×{3}

= {1,3},{2,3},{3,3},{4,3}

• Again,

(A×Q)

= {1,2,3,4}×{5,7}

= {1,5},{2,5},{3,5},{4,5}{1,7},{2,7},{3,7},{4,7}

(A×R)∩(A×Q)

= {{1,3},{2,3},{3,3},{4,3}} ∩{{1,5},{2,5},{3,5},{4,5},{1,7},{2,7},{3,7},{4,7}}

= ∅

$\begin{gathered}\;\;\;\;\underline{\boxed{\tt{(A×R)∩(A×Q)=∅}}}\end{gathered}$

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$\boxed{3}$ We get from (1) that,

• A = {1,2,3,4}

The elements of A are :

∅,{1},{2},{3},{4},{1,2},{1,3},{1,4},{2,3},{2,4},{3,4},{1,2,3},{1,3,4},{2,3,4},{1,2,4},{1,2,3,4}

So, P(A) = ∅,{1},{2},{3},{4},{1,2},{1,3},{1,4},{2,3},{2,4},{3,4},{1,2,3},{1,3,4},{2,3,4},{1,2,4},{1,2,3,4}

Here, The element number of A = 4 = n (suppose)

The element number of P(A) = 16 = $2^4$= $2^n$

$\begin{gathered}\;\;\;\;\underline{\boxed{\tt{So,The\:element\:number\:of\:P(A)\:supports\:2^n}}}\end{gathered}$