Final answer:
The operation defined by mon = (m + n + mn) mod 12 for a set {1, 2, 3, 4, 5} can be explored by creating a table to show its values. Commutativity can be tested by comparing values across the table and the truth sets for given equations are the values that satisfy each equation under the operation.
Step-by-step explanation:
The operation defined on the set R of real numbers by mon = (m + n + mn) mod 12 is a function that combines two numbers and returns another number, under modulo 12 arithmetic. To investigate the properties of this operation within the set {1, 2, 3, 4, 5}, we can create a table where the first element of the operation, m, is represented by the rows, and the second element, n, by the columns. After performing the operation for every pair (m, n), the resulting table can help us explore the operation's commutativity and solve equations within that set.
Drawing a Table
To draw the operation table, simply calculate the value of mon for each combination of m and n from the set {1, 2, 3, 4, 5} and place the result in a grid. The columns and rows will each represent a number in the set, and the cell at the intersection of row m and column n will show the result of mon.
Testing Commutativity
To show that the operation is commutative on the set {1, 2, 3, 4, 5}, one must verify that for all m, n in the set, the equation mon = nom holds true. This can be observed in the table if the value at the intersection of row m and column n is the same as the value at the intersection of row n and column m.
Finding the Truth Set
The truth set for the equation i) mon = n contains all values of m for which the equation is true for a given n. For ii) 30n = 11, the truth set includes all values of n such that when n is plugged into the operation with m = 30, the result is 11 under modulo 12 arithmetic.