Computer Methods - Assignment Example

Manchester Metropolitan Department of Computing and Mathematics MA1261 Computing Methods: Test Revision P, Q and R are the sets P = {1, 2, 3, 4}, Q = {3, 4, 5, 6} and R = {3, 5, 7, 9}. a) Draw the Venn diagram containing P, Q and R. [4] b) List the elements in the sets P - (Q'R) and (P - Q)'(P - R). Are these two sets equal' Justify your answer. [6] c) List the elements in T = X ''Y when X = P - Q and Y = P - R. Assuming T is the universal set and S = {(x, y), x x L(x) ''A (x) iii) Not every lecturer is articulate. Restating, It is not in the case that every x in L(x) is articulate. It is not in the case that every x in L(x) and x is articulate. ('oex L(x)) ''A (x) Both are special cases of predicate logic, known as Existence Quantification of Predicative logic in which, both the statements are logically equal. The third statement can be restated as, Some lecturers are not articulate. For some x in L(x), x in not articulate. > x L(x) ''A (x) c) Truth table consists of the outputs in the binary format of True or False. To draw the table of the relation (P_Q) ''''(P'Q) Step 1: P implies Q (logical implication) not P or Q (material conditional) Produces, false value if the element of first operand is true and the second is false. P Q P _ Q T T T T F F F T T F F T Step 2: Logically equivalent (P if and only if Q) Produces true value if both are true or both or false P Q P ''Q T T T T F F F T F F F T Step 3: Negation Produces the value of true if false and produces false if true. P Q P ''Q '(P ''Q) T T T F Step 4: Logical conjunction (''- or (union)) Produces the value True for at least one true. P Q P _ Q P ''Q '(P ''Q) (P_Q) ''''(P'Q) T T T T F T T F F F T T F T T F T T F F T T F T The result of the statement is true for all conditions of P an Q and hence it is tautology. T F F T F T F T F F T F 3. a) Given that ''''{x, x'R, 0 ''x ''1} and f(x) is the mapping from ''to R whose graph is shown on the right determine which of the following statements is true and which is false. i) f (x) is a total function. ii) f (x) is an injection. iii) f (x) is a surjection. Justify your conclusion in each case. [6] b) Assuming g: R'R, g(x) = 3x - 6 and find i) g(2) ii) g-1(x) iii) g_ g [9] Solution: a) Question not enough(Need the function to solve) b) g: R ''R by g(x) = 3x - 6 i) g (2) = 3(2) - 6 = 6 - 6 = 0 (substituting value of x = 2 in the function) g(2) = 0 ii) g-1(x) Let y = g(x) y = 3x - 6 3x = y + 6 x = (y + 6) / 3 Substituting x by g-1(x) and y by x, g-1(x) = (x + 6) /3 iii) g _ g = g(3x - 6) = 3(3x - 6) - 6 = 9x - 18 - 6 = 9x - 24 Read More