Give an example where $A \subseteq B$ with $A \neq B,$ but $\left\langle A\right\rangle= \left\langle B\right\rangle.$

Question

I read book of Dummit and Foot Abstract algebra. I need some help with the following exercises(2.4).

Let $\left\langle H\right\rangle$ be subgroup of $G$ generated by a subset $H$

1. Prove that if $H$ is a subgroup of $G$ then $\left\langle H\right\rangle = H.$

2. Prove that if $A$ is a subset of $B$ then $\left\langle A\right\rangle\leq \left\langle B\right\rangle.$ Give an example where $A \subseteq B$ with $A \neq B,$ but $\left\langle A\right\rangle= \left\langle B\right\rangle.$

I can prove both statements. But I cannot give an example where $A \subseteq B$ with $A \neq B,$ but $\left\langle A\right\rangle= \left\langle B\right\rangle.$