Query on homomorphism. If we say that $H:A\rightarrow B$ is a homomorphism from **A** to **B** , does it mean that **A is homomorphic to B** or **B is homomorphic to A**?. Are the two statements actually different? What is meant by the **homomorphic image** of **A**. And is it possibel for set **A** to have a smaller cardinality than set **B** if there is a homomorphism from **A** to **B**?.

You never say that $A$ is homomorphic to $B$; it would be an uninteresting notion as any two groups(/rings/whatever you are considering) would then be homomorphic through the homomorphism mapping everything to the trivial element. They can be "isomorphic" though, and this is then a non-trivial notion.

The term "homomorphic image" just refers to the image under a given homomorphism $A \to B$ with the structure that comes with it.

Finally, take $A$ to be the trivial group (or again, whatever is relevant to you), take $B$ to be something non-trivial, and let $A \to B$ be the map mapping the trivial element to the trivial element. This is a homomorphism, yet $A$ has smaller cardinality than $B$.