Can you design xor gate using or gate?
No. OR is not functionally complete, so you can not use it to
derive any other logical expression. The reason for this is because
you can only construct the following expressions out of only OR
gates:
A OR B
A OR A
Because of the Idempotency theorem, A OR A simply reduces to A,
so we are left with A OR B, which we can not use to derive any
other logical circuits. At the very least, we would also need a NOT
gate.
This is why NOR and NAND are functionally complete: you can
derive a NOT gate by using A NAND A or A NOR A.