Question 2: Let, \(U=\{x:x\text{ is an integer and }0<x\leq150\}\), \(G=\{x:x=2^m\text{ for integer m and }0\leq m\leq 12 \}\) and \(H=\{x:x\text{ is a square}\}\).
\((i)\) \(\;\) List all elements of sets \(G\) and \(H\) in tabular form
Solution:
\(G=\{1,2,4,8,16,32,64,128\}\)
\(H=\{1,4,9,16,25,36,49,64,81,100,121,144\}\)
Explanation:
\(G=\{x:x=2^m\text{ for integer m and }0\leq m\leq 12 \}\)
We’re finding all powers of \(2\) such that the result is less than or equal to \(150\) because our universal set is upto \(150\).
\(2^0=1,\:\)\(2^1=2,\;\)\(2^2=4,\;\)\(2^3=8,\;\)\(2^4=16,\;\)\(2^5=32,\;\)\(2^6=64,\;\)\(2^7=128\;\)
\(H=\{x:x\text{ is a square}\}\).
We want all perfect squares from 1 to 150.
\(1^2=1,\;\)\(2^2=4,\;\)\(3^2=9,\;\)\(4^2=16,\;\)\(5^2=25,\;\)\(6^2=36,\;\)\(7^2=49,\;\)\(8^2=64,\;\)\(9^2=81,\;\)\(10^2=100,\;\)\(11^2=121,\;\)\(12^2=144\;\)
Tips for Students:
- Always look at the condition \(x \leq 150\) to know when to stop.
- Use tables or calculators to compute powers and squares.