For this assignment, answer the following questions and provide a detailed step$-$by$-$step explanation of calculations wherever necessary.

$1.$ Based on the concepts of basic logical laws $($rules of inference$)$ for propositions, that you learned this week, check the validity of the arguments. Show all steps and write the logical laws applied at each step $($wherever required$).$

A $\backslash$rightarrow B$,$ A v C$,$ C $\backslash$rightarrow $$ B $,$ D $\backslash$rightarrow B$,$ C $\backslash$Lambda D $\backslash$Rightarrow B $\backslash$Lambda $$ C

$2.$ Explain quantifiers and its types. Provide examples of propositions involving quantifiers in words and represent them symbolically. Discuss the truth of the predicates you used as examples. $($Ensure that you do not take the same examples discussed in the textbooks or the internet$).$