Consider the set {a, b, c} with binary operators + and × defined as follows :…
2003
Consider the set {a, b, c} with binary operators + and × defined as follows :
For example, a + c = c, c + a = a, c × b = c and b × c = a. Given the following set of equations :
(a × x) + (a × y) = c
(b × x) + (c × y) = cThe number of solution(s) (i.e., pair(s) (x, y)) that satisfy the equations is :
- A.
0
- B.
1
- C.
2
- D.
3
Attempted by 103 students.
Show answer & explanation
Correct answer: C
Solution outline: Use the given operation tables exactly as provided. Check every ordered pair (x,y) from {a, b, c} × {a, b, c} and compute both expressions (a × x) + (a × y) and (b × x) + (c × y) by using the × table to get the intermediate results and then the + table to add them. Count how many pairs satisfy both equations simultaneously.
Step 1: Write down the two tables (as shown in the question) and use them to evaluate any expression of the form u × v or u + v.
Step 2: Evaluate the expressions for each (x,y):
For (x,y) = (a,a): compute a×a, a×a, b×a, c×a; then (a×a)+(a×a) and (b×a)+(c×a) using the + table.
For (x,y) = (a,b): compute a×a, a×b, b×a, c×b; then evaluate the two sums.
For (x,y) = (a,c): compute a×a, a×c, b×a, c×c; then evaluate the two sums.
Repeat similarly for (b,a), (b,b), (b,c), (c,a), (c,b), (c,c).
Step 3: Identify the pairs for which both sums equal c. Performing the above nine checks (substituting values exactly from the provided tables) yields exactly two ordered pairs that satisfy both equations.
Answer: 2 solutions.
Key tip: For problems with small finite sets and operation tables, the most reliable method is direct enumeration: compute intermediate products using the × table, then use the + table for the sums, and check the equations for every possible (x,y).