Answer all questions. Show all steps in the solution clearly. Please use clear handwritten.

Course: Discrete Mathematics

Question 1 (15 marks) i. Explain the relationship of domino effect with the mathematical Induction ii. Use Mathematical Induction to prove the following proposition. a ) 2(1) + 2(4) + 2(7) + ...+ 2(3n - 2) = 3n2 -n b) 2+22+ 23 + ... + 2" = 2(2" -1) C) 1(1!) + 2(2!) + 3(3!) + ...+ n(n!) = (n+1)-1 (2m- -1) =2" -n-1 d) m=1 [(3m' + m)= n(n+1)? e) m=1Question 2 (5 marks) Define your own set A with 7 numbers (example: Let A = {3,1,0,1,2,3,4}) ***each group should define different set of A ***Don't use Set A as in above example Let R be a binary relation on A such that R ={(x,y )| X + y is divisible by 2} Find the relation R. Determine the properties of R and state the reasons. Determine whether R is equivalence relation and state the reasons. If yes, find the equivalence class and partition. Determine whether R is partial order and state the reasons. Question 3 (5 marks) Let f ;= Z" R be a function and f(X) =7?. ***set your own function*** . . e Sketch the graph of f(x). **use any software/tool for sketching*** State the domain, codomain and the range of f(x). Determine whether the f(x) is one-to-one function and justify your answers Determine whether the f(x) is onto function and justify your answers Question 4 (5 marks) Example: |-27.14+21.49|+7 (trunc(25.78)) 248] Evaluate i. Create an expression consists of ceiling, floor and truncation and evaluate the function (sample as given above). i. Evaluate the expression. iii. Explain the purpose of ceiling, floor and truncation functions