Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 2 / 13I / b / i / Solution

The first assertion is false: $1<2$, but $1+\omega =2+\omega =\omega$. The second is true. In the concatenation definition of ordinal addition, the copy of $\alpha$ is a proper initial segment of the copy of $\beta$ when $\alpha <\beta$, so $\gamma +\alpha$ is a proper initial segment of $\gamma +\beta$.