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SubjectsMathematics
रिक्त समुच्चय से अनंत समुच्चय तक: चरण 3
This Class 10 Mathematics topic from the chapter Sets explains the concluding step used to show that a set is infinite. Students distinguish an empty set, which contains no elements, from an infinite set, whose elements do not end, and follow how a logical argument supports this conclusion. The topic strengthens understanding of set notation, membership, and mathematical reasoning, helping learners interpret why the existence of continually extendable elements rules out a finite count.
TOPIC PRACTICE
Up to 6 questions from this page. Select your focus, then start.
The braces in A = {0} show that 0 is an element of the set. There is exactly one distinct element, so A is a singleton set. A set containing one element is also finite. It is not empty, because the empty set contains no elements, and {0} is not equal to ∅. Therefore, option B is correct.
Find the natural-number solutions by factoring the quadratic: x² − 5x + 6 = (x − 2)(x − 3). Hence x = 2 or x = 3. Both values are natural numbers, so the set defined by the condition is A = {2, 3}. Therefore, option B is equal to A. Option A uses incorrect roots, option C gives unrelated negative values, and option D wrongly treats the solution set as empty.
The closed interval [1, 1] contains all real numbers x satisfying 1 ≤ x ≤ 1. The only real number that satisfies both inequalities is x = 1, and the square brackets indicate that the endpoint is included. Therefore [1, 1] contains exactly one element and is equal to the singleton set {1}. Option A is correct.
The open interval (1, 1) consists of real numbers x satisfying 1 < x < 1. No real number can be simultaneously greater than 1 and less than 1. The parentheses also exclude both endpoints, so 1 is not included. Consequently the interval contains no elements and is the empty set ∅. Hence option A is correct.
The symbol ≤ includes equality, so both boundary values are part of the set. The condition −1 ≤ x includes −1, and x ≤ 4 includes 4. In interval notation, an included endpoint is written with a square bracket. Since both endpoints are included, the set is the closed interval [−1, 4].
The complement of A ∪ B consists of all elements in the universal set U that are not in the union. For a finite universal set, n((A ∪ B)') = n(U) − n(A ∪ B). Substituting the values gives 60 − 45 = 15. Thus, option A is correct. In a Venn diagram, this is the area of the rectangle outside both circles.
QUIZ COMPLETE
