Question

If \(P = \{1, 2\}\), then \(P \times P \times P = \{(1,1,1), (1,1,2), (1,2,1), (2,1,2), (2,2,1), (2,2,2)\}\). The ordered triplet missing in \(P \times P \times P\) are:

• A. \((1,2), (2,1)\)
• B. \((1,2,1), (2,1,1)\)
• C. \((1,2,2), (2,1,2)\)
• D. \((1,2,2), (2,1,1)\)

Hint:

For Cartesian products, first calculate the total number of elements using (n^k), then systematically list all ordered tuples to identify any missing ones.

Answer 1

The Correct Option is D

Step 1: Determine the total number of elements in \(P \times P \times P\). If a set has \(n\) elements, then \(P \times P \times P\) has \(n^3\) elements. Here, \(P = \{1,2\}\) has \(2\) elements. \[ |P \times P \times P| = 2^3 = 8 \] Step 2: List all possible ordered triplets. All possible elements of \(P \times P \times P\) are: \[ (1,1,1), (1,1,2), (1,2,1), (1,2,2), \] \[ (2,1,1), (2,1,2), (2,2,1), (2,2,2) \] Step 3: Compare with the given set. Given elements: \[ (1,1,1), (1,1,2), (1,2,1), (2,1,2), (2,2,1), (2,2,2) \] Missing elements are: \[ (1,2,2) \quad \text{and} \quad (2,1,1) \] Step 4: Identify the correct option. Option (D) correctly lists the missing ordered triplets.