Question

pls need quick response solution
​

Answer 1

Given,

$A=\{x:x\ \text{is a factor of 12}\}$

And,

$B=\{x:x\ \text{is a factor of 6}\}$

We have to find  $A\cap B$  and  $A\cup B.$

We can find the answer without writing them in roster form.

We know the definition of union and intersection of two sets.

$A\cap B=\{x:x\in A\ \text{and}\ x\in B\}\\ \\ A\cup B=\{x:x\in A\ \text{or}\ x\in B\}$

So, in this case, we can say,

$A\cap B=\{x:x\ \text{is a factor of both 12 and 6}\}\\ \\ A\cup B=\{x:x\ \text{is a factor of either 12 or 6}\}$

But we see that 6 is a factor of 12. Hence probably it is true that  $B\subset A.$

Because, since 6 is a factor of 12, then all factors of 6 are those of 12 too.

So we may remember another thing.

$\text{Iff\ B\subset A, \ then\ A\cap B=B \ and\ A\cup B=A. }$

So we easily got the answer!

$(i)\ \ A\cap B=\bold{B}\\ \\ \Longrightarrow\ \{x:x\ \text{is a factor of both 12 and 6}\}=\{x:x\ \text{is a factor of 6}\}\\ \\ (ii)\ \ A\cup B=\bold{A}\\ \\ \Longrightarrow\ \{x:x\ \text{is a factor of either 12 or 6}\}=\{x:x\ \text{is a factor of 12}\}$

If needed, only A and B are written in roster form for the answer.

$A=\{1,\ 2,\ 3,\ 4,\ 6,\ 12\}\\ \\ B=\{1,\ 2,\ 3,\ 6\}$