If A = {x : x is a multiple of 7}, B = {x : x is a multiple of 5} and C = {x : x is a multiple of 35} Then which of the following is null set?
(A ∩ B) – C
Understanding Set Operations and Multiples
The problem involves understanding set operations like intersection, union, and difference, applied to sets defined by properties related to multiples of numbers. We are given three sets:
- Set A: \(A = \{x : x \text{ is a multiple of } 7\}\)
- Set B: \(B = \{x : x \text{ is a multiple of } 5\}\)
- Set C: \(C = \{x : x \text{ is a multiple of } 35\}\)
We need to determine which of the given expressions represents a null set (\(\emptyset\)), which is a set containing no elements.
Analyzing the Relationship Between Sets A, B, and C
Let's look at the intersection of sets A and B, denoted by \(A \cap B\). This set contains elements that are common to both A and B. An element \(x\) is in \(A \cap B\) if and only if \(x\) is a multiple of 7 AND \(x\) is a multiple of 5.
A number that is a multiple of both 7 and 5 must be a multiple of their least common multiple (LCM). The LCM of 7 and 5 is \(7 \times 5 = 35\).
Therefore, \(A \cap B = \{x : x \text{ is a multiple of } 35\}\).
Comparing this with the definition of set C, we see that \(C = \{x : x \text{ is a multiple of } 35\}\).
Thus, we have the important relationship: \(A \cap B = C\).
Evaluating Each Option Using Set Relationships
Now, let's evaluate each given expression using the relationship \(A \cap B = C\) and the definitions of set operations.
Option 1: \( (A - B) \cup C \)
- \(A - B\): This set contains elements that are in A but not in B. These are multiples of 7 that are not multiples of 5. For example, 7, 14, 21, 28, 42, etc. Multiples of 35 (like 35, 70) are multiples of both 7 and 5, so they are not in \(A - B\).
- \( (A - B) \cup C \): This is the union of the set of multiples of 7 (but not 5) and the set of multiples of 35. This set includes elements like 7 (from A-B) and 35 (from C). Since it contains elements, it is not the null set.
Option 2: \( (A - B) - C \)
- \(A - B\): As seen above, this set contains multiples of 7 that are not multiples of 5.
- \( (A - B) - C \): This set contains elements that are in \(A - B\) but not in C. Since \(A - B\) contains multiples of 7 that are NOT multiples of 5, none of its elements can be multiples of 35 (which are multiples of both 7 and 5). Therefore, the set C has no elements in common with \(A - B\). Subtracting C from \(A - B\) removes no elements. Thus, \( (A - B) - C = A - B \). Since \(A - B\) contains elements like 7, it is not the null set.
Option 3: \( (A \cap B) \cap C \)
- We know that \(A \cap B = C\).
- So, \( (A \cap B) \cap C = C \cap C \).
- The intersection of a set with itself is the set itself. \( C \cap C = C \).
- Set C is the set of multiples of 35, which is not a null set (e.g., 35 is in C).
Option 4: \( (A \cap B) - C \)
- We know that \(A \cap B = C\).
- So, \( (A \cap B) - C = C - C \).
- The difference of a set with itself, \(C - C\), contains elements that are in C and NOT in C. There are no such elements.
- Therefore, \( C - C = \emptyset \), the null set.
Based on our analysis, the expression \( (A \cap B) - C \) represents the null set.
Summary of Set Operations Results
| Expression | Evaluation using \(A \cap B = C\) | Result | Is it a Null Set? |
|---|---|---|---|
| \( (A - B) \cup C \) | \( (A - B) \cup (A \cap B) \) | Not null (e.g., contains 7, 35) | No |
| \( (A - B) - C \) | \( (A - B) - (A \cap B) \) which simplifies to \(A - B\) | Not null (e.g., contains 7) | No |
| \( (A \cap B) \cap C \) | \( C \cap C = C \) | Not null (e.g., contains 35) | No |
| \( (A \cap B) - C \) | \( C - C \) | \( \emptyset \) | Yes |
Revision Table: Key Set Theory Concepts
| Concept | Notation | Definition | Example |
|---|---|---|---|
| Intersection | \(A \cap B\) | Elements common to both A and B. \(x \in A \cap B \iff x \in A \text{ and } x \in B\) | If A={1,2,3}, B={2,3,4}, then \(A \cap B\)= {2,3} |
| Union | \(A \cup B\) | Elements in A or in B (or both). \(x \in A \cup B \iff x \in A \text{ or } x \in B\) | If A={1,2,3}, B={2,3,4}, then \(A \cup B\)= {1,2,3,4} |
| Set Difference | \(A - B\) or \(A \setminus B\) | Elements in A but not in B. \(x \in A - B \iff x \in A \text{ and } x \notin B\) | If A={1,2,3}, B={2,3,4}, then \(A - B\)= {1} |
| Null Set (Empty Set) | \( \emptyset \) or {} | A set containing no elements. | The set of prime numbers between 10 and 11 is \( \emptyset \) |
Additional Information on Set Theory Problems
When solving set theory problems involving properties like multiples, it's often helpful to:
- Write out the first few elements of each set to get a better understanding.
- Determine the relationship between the sets, especially involving intersection (LCM for multiples) and union.
- Use Venn diagrams to visualize the sets and the results of operations, although it was not strictly necessary for this particular problem once the \(A \cap B = C\) relationship was identified.
- Break down complex expressions step-by-step using the definitions of the set operations.
- Remember that the difference \(X - Y\) contains elements ONLY in X and NOT in Y. If Y is a subset of X, then \(X - Y\) contains elements in X that are not in Y. If Y is equal to X, \(X - Y\) is the null set.
Understanding these fundamental concepts and techniques is crucial for solving set theory problems efficiently.
Important Questions from Sets
The Cartesian product A × A has 16 elements among which are (0, 2) and (1, 3). Which of the following statements is/are correct?
1. It is possible to determine set A.
2. A × A contains the element (3, 2).
Select the correct answer using the code given below:
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Let $f(x) = |x - 2| + |x - 8|$; $x \in R$. Then the set of all values of $x$, at which the function, $g(x) = f(f(x))$ is not differentiable, is:
Consider the following statements in respect of two non-empty sets A and B :
1. A ∪ B = A ∩ B if A = B
2. A Δ B = ϕ if A = B
Which of the above statements is/are correct ?
