Explain the following with examples: $(a)$ $\alpha$ and $\beta$-hydrogen,$(b)$ $\beta$-elimination reaction,and $(c)$ Saytzeff rule.

(N/A) $\alpha, \beta$-Hydrogen: The carbon atom to which the halogen is attached is called the $\alpha$-carbon atom. The carbon atoms adjacent to this $\alpha$-carbon are called $\beta$-carbon atoms. The hydrogen atoms attached to the $\beta$-carbon are called $\beta$-hydrogen atoms.
$(b)$ Dehydrohalogenation or $\beta$-elimination reaction: When a haloalkane containing $\beta$-hydrogen is heated with an alcoholic solution of potassium hydroxide,a hydrogen atom is eliminated from the $\beta$-carbon and a halogen atom $(X)$ is eliminated from the $\alpha$-carbon,resulting in the formation of an alkene and a hydrogen halide $(HX)$. This reaction is generally known as a $\beta$-elimination or dehydrohalogenation reaction.
$(c)$ Saytzeff rule: In $1875$,the Russian chemist Alexander Saytzeff formulated a rule to predict the major alkene product formed from haloalkanes having more than one type of $\beta$-hydrogen. The rule states that in dehydrohalogenation reactions,the preferred product is the alkene that is more highly substituted (i.e.,the more stable alkene). For example,the reaction of $2$-bromobutane with alcoholic $KOH$ yields but-$2$-ene as the major product.