Question

Find the pairs of equal sets, if any, give reasons:
$A=\left\{0\right\},B=\{x:x>15\text{and}x<5\}$,
$C=\{x:x-5=0\},D=\{x:x^2=25\}$
$E=\{x:x$ is an integral positive root of the equation of $x^2-2x-15=0\}$

Answer 1

Step $1$ Solve for $A$
$A=\left\{0\right\}$

Step $2:$ Solve for $B$
$B=\{x:x>15\text{and}x<5\}$
No number can be greater than 15 and less than 5
simultaneously
$\therefore B=\varphi$

Step $3:$ Solve for $C$
$c=\{x:x-5=0\}$
$x-5=0$
$\Rightarrow x=5$
$\therefore C=\left\{5\right\}$

Step $4:$ Solve for $D$
$D=\{x:x^2=25\}$
$x^2=25$
$\Rightarrow x\pm 5$
$D=\{-5,5\}$

Step $5:$ Solve for $E$
$E=\{x:x$ is an integral positive root of the equation $x^2-2x-15=0\}$
$x^2-2x-15=0$
$\Rightarrow x^2-5x+3x-15=0$
$\Rightarrow (x-5)(x+3)=0\Rightarrow x=-3,5$
But $x$ is positive $\Rightarrow E=\left\{5\right\}$

Step $6:$ Comparing all sets

$A=\left\{0\right\},B=\varphi ,C=\left\{5\right\},D=\{-5,5\},E=\left\{5\right\}$

$\therefore$ Only $C=E,$ as their elements are the same.