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Let A={x1,x2,......Xm},B={y1,y2,.....,yn...

Let `A={x_1,x_2,......X_m},B={y_1,y_2,.....,y_n}` then total number of non-empty relations that can bedefined from A to B, is (i) `m^(n)` (ii) n^(m)-1`(iii) mn-1 (iv) `2^(mn)-1`

A

`m^(n)`

B

`n^(m)-1`

C

mn-1

D

`2^(mn)-1`

Text Solution

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The correct Answer is:
To solve the problem, we need to determine the total number of non-empty relations that can be defined from set A to set B. ### Step-by-Step Solution: 1. **Understanding Relations**: A relation from set A to set B is a subset of the Cartesian product A × B. If A has m elements and B has n elements, then the Cartesian product A × B will have mn elements. 2. **Total Subsets**: The total number of subsets of a set with k elements is given by \(2^k\). Therefore, the total number of subsets of the Cartesian product A × B, which has mn elements, is \(2^{mn}\). 3. **Non-Empty Relations**: Among these subsets, one of them is the empty relation (the subset that contains no pairs). To find the number of non-empty relations, we need to subtract this one empty relation from the total number of relations. Thus, the number of non-empty relations is: \[ 2^{mn} - 1 \] 4. **Conclusion**: Therefore, the total number of non-empty relations that can be defined from A to B is \(2^{mn} - 1\). ### Final Answer: The correct option is (iv) \(2^{mn} - 1\). ---
To solve the problem, we need to determine the total number of non-empty relations that can be defined from set A to set B. ### Step-by-Step Solution: 1. **Understanding Relations**: A relation from set A to set B is a subset of the Cartesian product A × B. If A has m elements and B has n elements, then the Cartesian product A × B will have mn elements. 2. **Total Subsets**: ...
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