Consider the following statements: 1. A = {1, 3, 5} and B = {2, 4, 7} are equivalent sets. 2. A = {1, 5, 9} and B = {1, 5, 5, 9, 9} are equal sets. Which of the above statements is/are correct?
Both 1 and 2
Analyzing Sets: Equivalent Sets vs. Equal Sets
This question asks us to evaluate two statements about sets. We need to understand the definitions of equivalent sets and equal sets to determine if the statements are correct.
Understanding Equivalent Sets
Two sets are considered equivalent sets if they have the same number of elements. The number of elements in a set is called its cardinality.
- If set A has cardinality $|A|$ and set B has cardinality $|B|$, then A and B are equivalent sets if and only if $|A| = |B|$.
Evaluating Statement 1: Equivalent Sets
Statement 1 says that A = {1, 3, 5} and B = {2, 4, 7} are equivalent sets.
Let's find the cardinality of each set:
- Set A = {1, 3, 5}. The elements are 1, 3, and 5. There are 3 elements. So, the cardinality of A is $|A| = 3$.
- Set B = {2, 4, 7}. The elements are 2, 4, and 7. There are 3 elements. So, the cardinality of B is $|B| = 3$.
Since $|A| = 3$ and $|B| = 3$, we have $|A| = |B|$. Therefore, set A and set B are equivalent sets.
Statement 1 is correct.
Understanding Equal Sets
Two sets are considered equal sets if they contain exactly the same elements. The order of elements does not matter, and repeating elements within the set notation does not change the set itself.
- If set A and set B contain precisely the same members, then A and B are equal sets.
- For example, {1, 2, 3} is equal to {3, 1, 2} and also equal to {1, 1, 2, 3, 3}.
Evaluating Statement 2: Equal Sets
Statement 2 says that A = {1, 5, 9} and B = {1, 5, 5, 9, 9} are equal sets.
Let's look at the elements in each set, considering that repetition of elements does not change the set:
- Set A = {1, 5, 9}. The distinct elements in set A are 1, 5, and 9.
- Set B = {1, 5, 5, 9, 9}. The distinct elements in set B are 1, 5, and 9. The repeated elements (5 and 9) are listed more than once, but they are still just the elements 5 and 9 within the set. So, set B is the same as {1, 5, 9}.
Since set A contains the elements {1, 5, 9} and set B also contains the elements {1, 5, 9} (when considering distinct elements), set A and set B are equal sets.
Statement 2 is correct.
Conclusion
Based on our analysis, both Statement 1 (A and B are equivalent sets) and Statement 2 (A and B are equal sets) are correct.
| Statement | Sets Given | Concept | Analysis | Conclusion |
|---|---|---|---|---|
| Statement 1 | A = {1, 3, 5} B = {2, 4, 7} |
Equivalent Sets (Same Cardinality) |
$|A| = 3$ $|B| = 3$ $|A| = |B|$ |
Correct |
| Statement 2 | A = {1, 5, 9} B = {1, 5, 5, 9, 9} |
Equal Sets (Same Elements) |
Distinct elements in A: {1, 5, 9} Distinct elements in B: {1, 5, 9} Elements are identical. |
Correct |
Revision Table: Set Concepts
| Concept | Definition | Condition | Example |
|---|---|---|---|
| Equivalent Sets | Sets that have the same number of elements (cardinality). | $|A| = |B|$ | A = {a, b} B = {1, 2} ($|A|=2, |B|=2$) |
| Equal Sets | Sets that contain exactly the same elements. | Every element of A is in B, and every element of B is in A. | A = {1, 2, 3} B = {3, 1, 2} (Same elements {1, 2, 3}) |
| Subset | Set A is a subset of B if every element in A is also in B. | If $x \in A$, then $x \in B$. Notation: $A \subseteq B$ | A = {1, 2} B = {1, 2, 3} (A is a subset of B) |
| Proper Subset | Set A is a proper subset of B if A is a subset of B and A is not equal to B. | $A \subseteq B$ and $A \ne B$. Notation: $A \subset B$ | A = {1, 2} B = {1, 2, 3} (A is a proper subset of B) |
Additional Information on Set Properties
Understanding basic set theory is fundamental in mathematics. Here are a few additional points about sets:
- Elements of a Set: The objects contained within a set are called its elements or members. We use the symbol $\in$ to denote membership (e.g., $x \in A$ means element $x$ is in set A) and $\notin$ for non-membership.
- Cardinality: The number of distinct elements in a finite set A is denoted by $|A|$ or $n(A)$.
- Empty Set: The empty set, denoted by $\emptyset$ or {}, is a unique set that contains no elements. Its cardinality is 0, i.e., $|\emptyset| = 0$. The empty set is a subset of every set.
- Universal Set: In a particular context, the universal set (usually denoted by U) is the set containing all possible elements relevant to that context.
- Equality Property: The key characteristic of equal sets is that they have exactly the same elements. The order in which the elements are listed and any repetition of elements do not affect the equality of sets. {a, b, c}, {c, a, b}, and {a, a, b, c, c} are all considered the same set.
- Relationship between Equal and Equivalent Sets: If two sets are equal, they must also be equivalent because they contain the exact same elements, thus the same number of elements. However, if two sets are equivalent, they are not necessarily equal (as shown in Statement 1 where {1, 3, 5} and {2, 4, 7} are equivalent but clearly not equal).
Important Questions from Set Theory and types of Sets
A set S contains (2n + 1) elements. There are 4096 subsets of S which contain at most n elements. What is n equal to?
Let A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Then the number of subsets of A containing exactly two elements is
If A = { x : x is a multiple of 3} and B = (x : x is a multiple of 4} and C = {x : x is a multiple of 12}, then which one of the following is a null set?
Let S be a set of all distinct numbers of the form \(\frac{{\rm{p}}}{{\rm{q}}}\) , where p, q ∈ {1, 2, 3, 4, 5, 6}. What is the the cardinality of the set S?
If A and B are two sets containing 2 elements and 4 elements respectively, then number of subsets of A × B having 3 or more elements is :
