To show equal cardinality To show two sets have equal cardinality is it always necessary to show bijection between the sets? 

suppose I have two sets A and B isn't it enough to show that there exist one onto mapping from A to B and some another onto mapping form B to A. The reason why I am asking this question is that it sometimes becomes very difficult to construct such a function which gives bijection.

For example, when I wanted to show bijection between rationals and naturals.
It was difficult to construct such a function and later found out that
$f(m,n)=2^{m-1}(2n-1)$

will do the work.

Now,recently I came across another problem that show that [0,1] and (0,1) have equal cardinality.
I was able to construct onto mapping from (0,1) to [0,1] and also another onto mapping from [0,1] to (0,1) but still now I couldn't construct such a function which will biject the two sets.
It isn't necessary to construct an explicit bijection if you have a result at hand you can apply. For instance, the Schroeder-Bernstein Theorem states that if there are injections $f : A \to B$ and $g : B \to A$, then there exists a bijection $h : A \to B$.

In your last example, it isn't hard to find injections $f : [0,1] \to (0,1)$ and $g : (0,1) \to [0,1]$, so there must exist a bijection between the sets, although we didn't demonstrate a specific example of one.
Existence of onto map from $A$ to $B$ gives card$(A)\leq$ card$(B)$ and existence of onto map from $B$ to $A$ gives card$(B)\leq$ card$(A)$ . In either way we can say that card$(A)=$ card$(B)$  i.e. there must be a bijection between $A$ and $B.$
But generally we have to show that two sets are of equal cardinality or not, we are not intersting in bijective map and thats  not so simple to find such a bijection.