Let A = {x € N: 3 ≤ x ≤ 13}, B = {x € N : x is even}, and C = {x EN: x is odd}. (a) Find An B. (b) Find A UB. (c) Find Bn C.. (d) Find BUC.

We are given that,

$A = {x \in \N: 3 ≤ x ≤ 13}, B = {x \in \N : x\space is \space even}, and\space C = {x \in \N: x\space is\space odd}.$  

then,

$A=\{3,4,5,6,7,8,9,10,11,12,13\}\\B=\{2,4,6,8,...\}\\C=\{1,3,5,7,9,......\}$

$a)$

$A\cap B$ consist of the elements common to both set $A$ and set $B$.

Therefore,

$A\cap B=\{4,6,8,10,12\}$  

$b)$

$A\cup B$ consist of all elements in set $A$ and set $B$.

Therefore,

$A\cup B=\{2,3,4,5,6,7,8,9,10,11,12,13, 14,16,18,20,.....\}$

For this case, $A\cup B$ consist of all even numbers and the odd numbers, 3,5,7,9,11,13 in set $A$.

$c)$

$B\cap C$  consist of elements common to both set B and set C. Clearly, $B\cap C=\{\empty\}.$ This is because, no element is common to both sets as set B consist of all even numbers and set C consist of all odd numbers. A number can either be even or odd.

$d)$

$B\cup C$ consist of all elements in set B and set C.

Therefore,

$B\cup C=\{1,2,3,4,5,6,7,8,9,10,11,12,13,...\}$. We can observe that this is the set  of natural numbers$(\N)$ .