1 Show that if ABC and D are sets with AB and CD then AXBBXD

1. Show that if A,B,C, and D are sets with |A|=|B| and |C|=|D|, then |AXB|=|BXD|.

2. Prove that a natural number X is divisible by 6 if and only if x is divisible by both 2 and 3.

1.Its given that the number of elements of set A=number of elements of B let |A|=|B|=n

and |C|=|D|=m where m and n are distinct natural numbers.

|AXB|=n² and |BXD|=nm assuming all the elements of set A ,B ,C ,D are disctinct

then |AXB| can be equal to |BXD| if m=n that is the number of elements of all sets are equal

2.Let a natural number divisible by 6 =6k(where k is a natural number)

now check if its divisible by 2, we can write 6k as 2*3k so it is divisible by 2 6k/2=3k=natural number

Ans we can say it is also divisible by 3 6k/3=2k=natural number

if a number is divisible by 2 and 3 it has to be 2k₁ an 3k₂ where k₁ and k₂ are natural numbers respetively.

but if divisible by both 2 and 3 it should 2*3C=6C where C is a natural number

and 6C is divisible by 6 6C/6=C= natural number