Let `A = { x,y,z}` and B `= { p,q,r,s}` . What is the number distinct relations from B to A ?

Options

A. 4096
B. 4094
C. 128
D. 126

Answer

To find the number of distinct relations from set B to set A, we can follow these steps:

### Step 1: Understand the Concept of Relations
A relation from set B to set A is a subset of the Cartesian product \( B \times A \). The Cartesian product consists of all possible ordered pairs where the first element is from set B and the second element is from set A.

### Step 2: Determine the Sizes of the Sets
Identify the number of elements in each set:
- Set A = {x, y, z} has 3 elements.
- Set B = {p, q, r, s} has 4 elements.

### Step 3: Calculate the Size of the Cartesian Product
The size of the Cartesian product \( B \times A \) is given by multiplying the number of elements in set B by the number of elements in set A:
\[ |B \times A| = |B| \times |A| = 4 \times 3 = 12 \]

### Step 4: Calculate the Number of Distinct Relations
The number of distinct relations from set B to set A is equal to the number of subsets of the Cartesian product \( B \times A \). Since each element of \( B \times A \) can either be included in a relation or not, the total number of distinct relations is given by:
\[ \text{Number of distinct relations} = 2^{|B \times A|} = 2^{12} \]

### Step 5: Calculate \( 2^{12} \)
Now we can compute \( 2^{12} \):
\[ 2^{12} = 4096 \]

### Conclusion
Thus, the total number of distinct relations from set B to set A is \( 4096 \).