Problem 61 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(\mathrm{A}\) and \(\mathrm{B}\) two sets containing 2 elements and 4 elements respectively. The number of subsets of \(\mathrm{A} \times \mathrm{B}\) having 3 or more elements is (a) 256 (b) 220 (c) 219 (d) 211

Step-by-Step Solution

Answer

The number of subsets of \(A \times B\) having 3 or more elements is 219.

Step 1: Determine the cardinality of A × B

First, calculate the number of elements in the set \(A \times B\). Since set \(A\) has 2 elements and set \(B\) has 4 elements, the product \(A \times B\) will contain \(2 \times 4 = 8\) elements.

Step 2: Calculate the total number of subsets

The total number of subsets of a set with \(n\) elements is given by \(2^n\). Therefore, the number of subsets for \(A \times B\), which has 8 elements, is \(2^8 = 256\).

Step 3: Calculate the number of subsets with fewer than 3 elements

Subsets with fewer than 3 elements can be:- Subsets with 0 elements, which is 1 subset (the empty set).- Subsets with 1 element, which can be formed in \(\binom{8}{1} = 8\) ways.- Subsets with 2 elements, which can be formed in \(\binom{8}{2} = 28\) ways.Adding these, the number of subsets with fewer than 3 elements is \(1 + 8 + 28 = 37\).

Step 4: Calculate the number of subsets with 3 or more elements

To find the subsets with 3 or more elements, subtract the result from Step 3 from Step 2. Thus, the number of subsets with 3 or more elements is \(256 - 37 = 219\).

Key Concepts

Cartesian Product

The concept of the Cartesian product is essential in set theory. It involves combining two sets to form a set of ordered pairs. To understand it better, imagine two sets, \(A\) with \(a\) elements and \(B\) with \(b\) elements. The Cartesian product \(A \times B\) consists of all possible pairs \((x, y)\) where \(x\) is an element from \(A\) and \(y\) is from \(B\). This results in \(a \times b\) pairs in total, as each element of \(A\) pairs with each element of \(B\). For example, if \(A = \{1, 2\}\) and \(B = \{x, y, z, w\}\), the Cartesian product \(A \times B\) produces pairs like \((1, x), (1, y)\), and so on, for a total of \(2 \times 4 = 8\) pairs. The process illustrates how Cartesian products expand the number of elements, which subsequently affects calculations involving these sets, such as determining subsets.

Subset

Subsets are fundamental components of set theory. A subset is a collection derived from a larger set where all elements in the subset are also in the original set. If you consider a set \(X\), a subset \(Y\) would mean every element of \(Y\) is also an element of \(X\). There are varying sizes of subsets ranging from the empty set \(\emptyset\) (which contains no elements and is a subset of every set) to the set itself. In any set with \(n\) elements, there are \(2^n\) different subsets, including the empty set and the set itself. For example, if we have set \(D = \{a, b, c\}\), its subsets include \(\emptyset\), \(\{a\}\), \(\{a, b\}\), and \(\{a, b, c\}\) among others. In exercises dealing with subsets, it's crucial to understand not only how subsets are structured but also how many there are, which involves using the formula \(2^n\). This is especially pertinent in problems like the original exercise where we calculate subsets of a Cartesian product.

Combinatorics

Combinatorics is the field of mathematics that focuses on counting, arrangement, and combination of elements within a set. It is heavily used in problems involving subsets and permutations, just like in the given exercise. A frequent task in combinatorics is determining the number of ways to choose a subset from a larger set, often calculated using binomial coefficients \(\binom{n}{k}\), which denote choosing \(k\) elements from \(n\) elements without regard to the order. Formula:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\] In our original problem, we needed to find the number of subsets of a Cartesian product that contain 3 or more elements. This involved using combinatorial methods to determine how many subsets exist overall \((2^n)\) and subtracting the number that do not meet the criteria (having fewer than 3 elements). Understanding combinatorics allows you to efficiently navigate through these calculations and solve complex problems involving combinations and permutations in set theory.