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21h
answered Prove by induction: A tree on n≥2 vertices has ≥2 leaves
1d
comment Prove by induction: A tree on n≥2 vertices has ≥2 leaves
This proof is fine if you already know that every tree on $k+1$ vertices can be obtained by adding a vertex to some tree on $k$ vertices. Just how do you know that?
1d
comment Combinations choosing equal groups results in division by 2
Do you think $6$ is the right answer to the last problem in my common? If you have $3$ people, call them $A,B,C,$ there are $6$ ways to choose two teams of $1$? OK, let's list them.1. A&B 2. A&C 3. B&A. 4. B&C. 5. C&A. 6. C&B. The reason I divided by two is that I didn't think A&B and B&A should be counted as two different ways of making two teams of one. Of course, if they are to be counted as different ways, then I don't divide by $2$ and the answer if $6.$
1d
comment Combinations choosing equal groups results in division by 2
Sometimes looking at a smaller example makes it easier to see what is going on. $3$ people, split into team of $2$ and team of $1,$ the number of ways is $\binom32\binom11=3\cdot1=3.$ Split them into two teams of $1,$ the number of ways is $\binom31\binom21/2=3\cdot2/2=3.$ Why?
1d
revised Partition of natural number not equal to factorial
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1d
answered Partition of natural number not equal to factorial
2d
comment Existence of how many sets is asserted by the axiom of choice in this case?
What do you mean, "letting them all exist has its own problems"? It seems to be a necessary truth that they all exist, axiom of choice or no axiom of choice.
2d
revised binomial coefficient where k > n
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2d
answered binomial coefficient where k > n
Apr
28
comment What book about algebraic combinatorics is it?
Why are you asking us instead of Dr. Balogh?
Apr
28
comment Infinitely countable subset of $\mathbb{R}^2$ is connected.
Or just draw a continuum of lines, each intersecting the segment aa' and perpendicular to it. Pick one of those parallel lines which misses the set $A.$
Apr
27
revised How many distinct patterns exist for a 5x5 grid by filling 3 colors?
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Apr
27
comment Minimum number of marked squares on $n × n$ board
You only have to mark every 3rd row, right? So for a 6 by 6 board, you only have to make the squares in rows 2 and 5. If n is divisible by 3, then you can get by with marking $n^2/3$ instead of $n^2/2$ squares.
Apr
27
comment Simple connected graph question
"Simple" is irrelevant. If the graph has the minimum number of edges for connectivity, it won't have two vertices joined by two edges.
Apr
27
comment How can we show that the adjacency matrix of a regular graph commutes with its complement
What is the complement of a matrix??
Apr
26
revised How do I write $y''+y' +\sin y \cos y = 0$ as a first order system?
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Apr
26
answered How do I write $y''+y' +\sin y \cos y = 0$ as a first order system?
Apr
25
answered Suppose $d_1$ and $d_2$ are equivalent metrics and $d_1$ is bounded, is $d_2$ bounded?
Apr
25
revised Suppose $d_1$ and $d_2$ are equivalent metrics and $d_1$ is bounded, is $d_2$ bounded?
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Apr
25
comment Suppose $d_1$ and $d_2$ are equivalent metrics and $d_1$ is bounded, is $d_2$ bounded?
If $d(x,y)$ is a metric, doesn't the metric $d_1(x,y)=\min\{1,d(x,y)\}$ induce the same topology as $d(x,y)$?