Problem 32 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(A\) and \(B\) be two sets containing 2 elements and 4 elements respectively. The number of subsets of \(A \times B\) having 3 or more elements is (A) 220 (B) 219 (C) 211 (D) 256

Step-by-Step Solution

Answer

219 subsets (option B).

Step 1: Understand the Concept of Cartesian Product

The Cartesian product of two sets \(A\) and \(B\), denoted as \(A \times B\), is the set of all ordered pairs \((a, b)\) where \(a\) is in \(A\) and \(b\) is in \(B\). If \(|A| = 2\) and \(|B| = 4\), then \(|A \times B| = 2 \times 4 = 8\).

Step 2: Calculate the Total Number of Subsets

To find the total number of subsets of a set, use the formula \(2^n\), where \(n\) is the number of elements in the set. For \(A \times B\), there are 8 elements, so the total number of subsets is \(2^8 = 256\).

Step 3: Determine Subsets with Fewer than 3 Elements

We need to consider the number of subsets with 0, 1, and 2 elements. A subset with 0 elements is the empty set, and there is exactly 1 empty set. The number of subsets with 1 element is 8, as each element can be chosen separately. The number of subsets with exactly 2 elements involves choosing 2 elements out of 8, calculated as \(\binom{8}{2} = \frac{8 \times 7}{2 \times 1} = 28\). Thus, the total subsets with fewer than 3 elements are \(1 + 8 + 28 = 37\).

Step 4: Calculate Subsets with 3 or More Elements

To find the number of subsets with 3 or more elements, subtract the number of subsets with fewer than 3 elements from the total number of subsets. This gives us \(256 - 37 = 219\).

Step 5: Pick the Correct Answer

From the above calculations, the correct option for the number of subsets of \(A \times B\) with 3 or more elements is \(219\), so the answer is option (B).

Key Concepts

Cartesian Product

In set theory, the Cartesian product of two sets, denoted as \(A \times B\), is a fundamental concept that involves pairing elements of each set to form ordered pairs. For instance, if you have two sets \(A = \{a, b\}\) and \(B = \{1, 2, 3, 4\}\), the Cartesian product \(A \times B\) would consist of all possible ordered pairs where the first component is from \(A\) and the second is from \(B\).

• Each element of \(A\) can pair with each element of \(B\).
• Therefore, the number of elements in \(A \times B\) is the product of the number of elements in \(A\) and \(B\).

This means \(|A \times B| = 2 \times 4 = 8\) because \(A\) has 2 elements and \(B\) has 4, resulting in 8 distinct ordered pairs.

Subsets

Subsets are variations of a set which can contain any number of elements from the original set, including none (the empty set) and the whole set itself. The concept is based on choosing different combinations of elements from a set.

• An important property is that a set with \(n\) elements has \(2^n\) subsets.
• This accounts for all possible combinations including the empty set and the full set.

For our example with a set \(A \times B\) containing 8 elements, there are \(2^8 = 256\) possible subsets. This includes subsets of all sizes, from the empty subset to the full set itself.

Combinatorics

Combinatorics is a branch of mathematics focusing on counting, arrangement, and combination of objects. It is especially useful in calculating the number of subsets with specific properties. In our problem, to calculate subsets with fewer than 3 elements from \(A \times B\), we apply combinatorial techniques.

• The empty set is trivial and there is exactly 1 subset with 0 elements.
• The number of subsets with exactly 1 element is equal to the number of elements, which is 8.
• To find subsets of exactly 2 elements from 8, use the combination formula \(\binom{8}{2}\), resulting in 28.

Hence, the subsets with fewer than three elements are counted as 1 (empty set) + 8 (one element) + 28 (two elements) = 37 subsets.

JEE Mathematics

The Joint Entrance Examination (JEE) is a highly competitive entrance exam for engineering colleges in India. Understanding set theory, particularly topics like the Cartesian product and subsets, is crucial for this exam as it often features in various permutations and combination problems.

• JEE aspirants must grasp these concepts to tackle questions involving subsets counting and combinations effectively.
• This problem, for instance, tests both basic knowledge of set operations and application of combinatorial principles.

By understanding how to calculate the number of subsets with certain characteristics, students can handle similar problems with confidence, improving their performance in exams like JEE.