Let A = {1, 2, 3, 4, 5, ..., 10}. The number of total relations on A is equal to 2^λ, then λ is equal to?
Understand the Problem
The question is asking about the number of total relations on a set A with 10 elements and how that relates to the expression 2 raised to the power of λ. We need to find the value of λ based on the given information about relations.
Answer
The value of \( \lambda \) is \( 100 \).
Answer for screen readers
The value of ( \lambda ) is ( 100 ).
Steps to Solve
- Understanding Relations on a Set
A relation on a set is a subset of the Cartesian product of that set with itself. For the set ( A ) with 10 elements, the Cartesian product ( A \times A ) has ( 10 \times 10 = 100 ) pairs.
- Calculating the Total Number of Subsets
The total number of relations is equal to the number of subsets of ( A \times A ). For a set with ( n ) elements, the number of subsets is ( 2^n ). Here, ( n = 100 ).
- Substituting into the Formula
Using the formula for subsets, the total number of relations on ( A ) is: $$ 2^{100} $$
- Finding ( \lambda )
Since the number of total relations on ( A ) is equal to ( 2^\lambda ): $$ 2^{100} = 2^\lambda $$
From this, we find that: $$ \lambda = 100 $$
The value of ( \lambda ) is ( 100 ).
More Information
In set theory, a relation is a way of linking elements from one set to another. The total number of relations on a set of size ( n ) grows exponentially, which highlights the vastness of combinations even in small sets.
Tips
- Confusing the number of relations with the number of elements in the set. The number of relations is based on pairs, not individual elements.
- Forgetting to calculate the total number of pairs in the Cartesian product.
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