RRB Technician Grade 1 Model Paper 1

Concept:The distributive law in Boolean algebra shows how AND and OR operations distribute over each other.Explanation:The distributive law has two standard forms:AND over OR: $\mathrm{A}(\mathrm{B} + \mathrm{C}) = \mathrm{AB} + \mathrm{AC}$OR over AND: $\mathrm{A} + \mathrm{BC} = (\mathrm{A} + \mathrm{B})(\mathrm{A} + \mathrm{C})$Option C represents the OR-over-AND form. Let us verify it.Start with $(\mathrm{A} + \mathrm{B})(\mathrm{A} + \mathrm{C})$.Expanding gives $\mathrm{AA} + \mathrm{AC} + \mathrm{AB} + \mathrm{BC}$.Using the idempotent law, $\mathrm{AA} = \mathrm{A}$, so we get:$\mathrm{A} + \mathrm{AC} + \mathrm{AB} + \mathrm{BC}$.Factor out $\mathrm{A}$:$\mathrm{A}(1 + \mathrm{C} + \mathrm{B}) + \mathrm{BC}$.Since $1 + \mathrm{C} + \mathrm{B} = 1$, the expression becomes:$\mathrm{A} + \mathrm{BC}$.Thus $(\mathrm{A} + \mathrm{B})(\mathrm{A} + \mathrm{C}) = \mathrm{A} + \mathrm{BC}$, confirming the distributive law.Answer:Option C: $\mathrm{A} + (\mathrm{BC}) = (\mathrm{A} + \mathrm{B})(\mathrm{A} + \mathrm{C})$.