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Excel in math and science
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Set Notation
When Cantor introduced his classification of multiple infinities, he was vehemently rejected by most mathematicians. Ye be warned: contemplating the continuum hypothesis can drive anyone a little mad!
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Concept Quizzes
If \(S=\{2,6,8\},\) which of the following is the power set of \(S?\)
A. \( \{ \{2\}, \{6\}, \{8\}, \{2,6\}, \{2,8\},\{6,8\}, \{2,6,8\} \} \)
B. \( \{ \emptyset, \{2\}, \{6\}, \{8\}, \{2,6\}, \{2,8\},\{6,8\}, \{2,6,8\} \} \)
C. \( \{ \emptyset, \{2\}, \{6\}, \{8\}, \{2,6\}, \{2,8\},\{6,8\} \} \)
D. \( \{ \{2\}, \{6\}, \{8\}, \{2,6\}, \{2,8\},\{6,8\} \} \)
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
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including olympiad champions, researchers, and professionals.
Let \[X=\{6, -\alpha\}, Y=\{\alpha^2+5,\alpha-8,1\}.\] If \(X \subseteq Y,\) what is the real number \(\alpha?\)
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
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examples, and problems from the community.
Used and loved by 4 million people
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Given the set \[A=\{x \mid x \text{ is a positive integer } < 13\},\] how many subsets of \(A\) have no even numbers?
Details and assumptions
The empty set \(\emptyset\) has no elements, and hence, has no even numbers.
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
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If \(A=\{5, 6, 7\}\) and \(B=\{6, 7, 9, 11, 15\},\) how many sets \(X\) are there satisfying
\[(A-B) \cup X = X, (A \cup B) \cap X = X?\]
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
If \(A\) is the set
\[A=\{5, 7, 11, 14\},\] how many subsets of \(A\) do not contain the element \(7?\)