Question

Consider two sets $A=\left( 2,4 \right)$ and$B=\left[ 3,5 \right)$ , find$A\cap B$.

Answer 1

Hint: For understanding the given $A=\left( 2,4 \right)$ and, we must first understand the meaning of different types of brackets. So, if a number is say$x$, following the condition $a < x < b$, then here $x$ can be all numbers from $a$ to$b$except $a$ and$b$ because there is no equal sign. It can also be represented as $x\in \left( a,b \right)$.

Complete step-by-step answer:

Here, the parentheses or round bracket$'\left( {} \right)'$ here is called an open bracket which means here that $a$ and $b$are not included. But if x is as follows, that is $a \le x < b$ , then it is represented as $x\in \left[ a,b \right)$ . Here before $a$ there is a square bracket which means $a$ is included and at $b$ there is a round bracket which means $b$is excluded.
So, let $x$ be a general element of $A$ ,
then $A=\left( 2,4 \right)$ means $A$ contains all elements greater than $2$ and less than $4$ other than $2$ and $4$ itself that is $2 < x < 4$ .
Also, let $y$ be the general element of $B$,
then $B=\left[ 3,5 \right)$means it contains all elements greater than equal to $3$ and less than $5$ excluding $5$that is $3 \le y < 5$ .
Now, $A\cap B$ means we have to find all numbers which is in$A$ as well as in $B$ that is common numbers. So, we have to find a $z$ such that $z$satisfies both ;
$2 < x < 4$ and $3\le y < 5$
So, in the number line we see,
$\begin{align}
  & \\
 & ~\underset{\begin{matrix}
   | & | & | & | & | & | & | & | & | & | & | \\
   0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\end{matrix}}{\longleftrightarrow} \\
\end{align}$


$z$ which is in both set$A$and set$B$ is $3\le z < 4$ which in set form is $z\in \left[ 3,4 \right)$ .
Here, $3$ is included as $3$is in set$A$ as well as in set$B$, but $4$ is excluded as it is only in set$B$ and not in set$A$. Hence,$A\cap B=\left[ 3,4 \right)$ .

Note: Brackets play a major role in these kind of questions .So, if in a question it is given as set$A=\left\{ 2,4 \right\}$,this means that set$A$ contain only two elements $2$and$4$,whereas if it is as$A=\left( 2,4 \right)$ this means set$A$ has all numbers greater than $2$ and less than$4$ excluding $2$ and $4$.If set$A=\left[ 2,4 \right]$, this means it has all numbers from $2$ to $4$ including $2$ and$4$.