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Excel in math and science
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Bijections
Excel in math and science
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Concept Quizzes
Consider the two sets \[X=\{1,2,3,4\}, Y=\{-8, 0, 20\}.\] Let \(f\) be a surjective function from \(X\) to \(Y\) such that for any two elements \(x_1\) and \(x_2\) of \(X,\) if \(x_1 < x_2,\) then \(f(x_1) \leq f(x_2).\) What is the minimum possible value of \(f(4)\)?
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.
A function \(f\) maps the elements of \(A = \{14, 16, 18, 20\}\) to elements of \(B =\{55, 66, 77, 88, 99\}.\) How many of the possible maps \(f\) are not injective?
Details and assumptions
A function is injective if each element in the codomain is mapped onto by at most one element in the domain.
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.
For two sets \[X=\{a,b,c\}, Y=\{7, 11, 13, 17, 25, 32\},\] \(f\) is an injective function from \(X\) to \(Y\). If \(f(a)=7\) and \(f(b)=17\), what is the sum of all the elements of \(Y\) that can possibly be the value of \(f(c)\)?
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.
For \(X=\{-2,-1,0,1,2\},\) function \(f\) is surjective from \(X\) to \(Y,\) where \[f(x) = \begin{cases} x+4 & \text{ if } x > 0, \\ -x^3+18 & \text{ if } x \leq 0. \end{cases} \] What is the sum of all the elements of \(Y?\)
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.
For two sets \[X=\{a,b,c\}, Y=\{y\mid 1 \leq y \leq 7, y \text{ is an integer}\},\] how many injective functions \(f: X \to Y\) exist?