Mixing
Sum and product notation
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I'm working with logic, but I need help with notation.
I'll give examples of what I want, because you will see the pattern. For each $n$, I want to perform "AND" on each pair, and OR all of the pairs together.
For $n = 1$, I want $a_{1}$
For $n = 2$, I want $(a_{1} wedge a_{2})$
For $n = 3$, I want $(a_{1} wedge a_{2}) vee (a_{1} wedge a_{3}) vee (a_{2} wedge a_{3})$
I want to write this with formal notation. I tried
$$bigvee_{i=1}^{n-1} (a_{i} wedge a_{i+1}), $$
but it doesn't work for $n = 3$. Any ideas? I think it might involve two AND/OR's, and I suspect that the second AND/OR will begin at the outside AND/OR's index.
notation index-notation
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add a comment |
$begingroup$
I'm working with logic, but I need help with notation.
I'll give examples of what I want, because you will see the pattern. For each $n$, I want to perform "AND" on each pair, and OR all of the pairs together.
For $n = 1$, I want $a_{1}$
For $n = 2$, I want $(a_{1} wedge a_{2})$
For $n = 3$, I want $(a_{1} wedge a_{2}) vee (a_{1} wedge a_{3}) vee (a_{2} wedge a_{3})$
I want to write this with formal notation. I tried
$$bigvee_{i=1}^{n-1} (a_{i} wedge a_{i+1}), $$
but it doesn't work for $n = 3$. Any ideas? I think it might involve two AND/OR's, and I suspect that the second AND/OR will begin at the outside AND/OR's index.
notation index-notation
$endgroup$
$begingroup$
For $n=3$, does the last term of your formula mean either $a_2 land a_3$ or $a_2 lor a_3$?
$endgroup$
– Doyun Nam
Jan 31 at 2:38
$begingroup$
it was a typo i fixed it
$endgroup$
– user614735
Jan 31 at 2:38
$begingroup$
This is perhaps not the most elegant, but my first instinct was to use $bigveelimits_{{i,j}inbinom{[n]}{2}}(a_iwedge a_j)$, using the notation that $[n]={1,2,3,dots,n}$ and $binom{A}{k}$ with $A$ a set is the set of subsets of size $k$ of $A$. This doesn't work for $n=1$, but should work for all larger $n$. To be fair, the meaning I use for the notation $binom{A}{k}$ is not widely used outside of smaller circles in combinatorics.
$endgroup$
– JMoravitz
Jan 31 at 2:40
add a comment |
$begingroup$
I'm working with logic, but I need help with notation.
I'll give examples of what I want, because you will see the pattern. For each $n$, I want to perform "AND" on each pair, and OR all of the pairs together.
For $n = 1$, I want $a_{1}$
For $n = 2$, I want $(a_{1} wedge a_{2})$
For $n = 3$, I want $(a_{1} wedge a_{2}) vee (a_{1} wedge a_{3}) vee (a_{2} wedge a_{3})$
I want to write this with formal notation. I tried
$$bigvee_{i=1}^{n-1} (a_{i} wedge a_{i+1}), $$
but it doesn't work for $n = 3$. Any ideas? I think it might involve two AND/OR's, and I suspect that the second AND/OR will begin at the outside AND/OR's index.
notation index-notation
$endgroup$
I'm working with logic, but I need help with notation.
I'll give examples of what I want, because you will see the pattern. For each $n$, I want to perform "AND" on each pair, and OR all of the pairs together.
For $n = 1$, I want $a_{1}$
For $n = 2$, I want $(a_{1} wedge a_{2})$
For $n = 3$, I want $(a_{1} wedge a_{2}) vee (a_{1} wedge a_{3}) vee (a_{2} wedge a_{3})$
I want to write this with formal notation. I tried
$$bigvee_{i=1}^{n-1} (a_{i} wedge a_{i+1}), $$
but it doesn't work for $n = 3$. Any ideas? I think it might involve two AND/OR's, and I suspect that the second AND/OR will begin at the outside AND/OR's index.
notation index-notation
notation index-notation
edited Jan 31 at 2:39
asked Jan 31 at 2:33
user614735
$begingroup$
For $n=3$, does the last term of your formula mean either $a_2 land a_3$ or $a_2 lor a_3$?
$endgroup$
– Doyun Nam
Jan 31 at 2:38
$begingroup$
it was a typo i fixed it
$endgroup$
– user614735
Jan 31 at 2:38
$begingroup$
This is perhaps not the most elegant, but my first instinct was to use $bigveelimits_{{i,j}inbinom{[n]}{2}}(a_iwedge a_j)$, using the notation that $[n]={1,2,3,dots,n}$ and $binom{A}{k}$ with $A$ a set is the set of subsets of size $k$ of $A$. This doesn't work for $n=1$, but should work for all larger $n$. To be fair, the meaning I use for the notation $binom{A}{k}$ is not widely used outside of smaller circles in combinatorics.
$endgroup$
– JMoravitz
Jan 31 at 2:40
add a comment |
$begingroup$
For $n=3$, does the last term of your formula mean either $a_2 land a_3$ or $a_2 lor a_3$?
$endgroup$
– Doyun Nam
Jan 31 at 2:38
$begingroup$
it was a typo i fixed it
$endgroup$
– user614735
Jan 31 at 2:38
$begingroup$
This is perhaps not the most elegant, but my first instinct was to use $bigveelimits_{{i,j}inbinom{[n]}{2}}(a_iwedge a_j)$, using the notation that $[n]={1,2,3,dots,n}$ and $binom{A}{k}$ with $A$ a set is the set of subsets of size $k$ of $A$. This doesn't work for $n=1$, but should work for all larger $n$. To be fair, the meaning I use for the notation $binom{A}{k}$ is not widely used outside of smaller circles in combinatorics.
$endgroup$
– JMoravitz
Jan 31 at 2:40
$begingroup$
For $n=3$, does the last term of your formula mean either $a_2 land a_3$ or $a_2 lor a_3$?
$endgroup$
– Doyun Nam
Jan 31 at 2:38
$begingroup$
For $n=3$, does the last term of your formula mean either $a_2 land a_3$ or $a_2 lor a_3$?
$endgroup$
– Doyun Nam
Jan 31 at 2:38
$begingroup$
it was a typo i fixed it
$endgroup$
– user614735
Jan 31 at 2:38
$begingroup$
it was a typo i fixed it
$endgroup$
– user614735
Jan 31 at 2:38
$begingroup$
This is perhaps not the most elegant, but my first instinct was to use $bigveelimits_{{i,j}inbinom{[n]}{2}}(a_iwedge a_j)$, using the notation that $[n]={1,2,3,dots,n}$ and $binom{A}{k}$ with $A$ a set is the set of subsets of size $k$ of $A$. This doesn't work for $n=1$, but should work for all larger $n$. To be fair, the meaning I use for the notation $binom{A}{k}$ is not widely used outside of smaller circles in combinatorics.
$endgroup$
– JMoravitz
Jan 31 at 2:40
$begingroup$
This is perhaps not the most elegant, but my first instinct was to use $bigveelimits_{{i,j}inbinom{[n]}{2}}(a_iwedge a_j)$, using the notation that $[n]={1,2,3,dots,n}$ and $binom{A}{k}$ with $A$ a set is the set of subsets of size $k$ of $A$. This doesn't work for $n=1$, but should work for all larger $n$. To be fair, the meaning I use for the notation $binom{A}{k}$ is not widely used outside of smaller circles in combinatorics.
$endgroup$
– JMoravitz
Jan 31 at 2:40
add a comment |
1 Answer
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For $n geq 2$,
$$bigveelimits_{j=2}^n big(bigveelimits_{i=1}^{j-1}(a_i land a_j) big)$$
might satisfy your formula.
answered Jan 31 at 2:55
Doyun NamDoyun Nam
66619
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add a comment |
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$begingroup$
For $n geq 2$,
$$bigveelimits_{j=2}^n big(bigveelimits_{i=1}^{j-1}(a_i land a_j) big)$$
might satisfy your formula.
answered Jan 31 at 2:55
Doyun NamDoyun Nam
66619
$endgroup$
add a comment |
$begingroup$
For $n geq 2$,
$$bigveelimits_{j=2}^n big(bigveelimits_{i=1}^{j-1}(a_i land a_j) big)$$
might satisfy your formula.
answered Jan 31 at 2:55
Doyun NamDoyun Nam
66619
$endgroup$
add a comment |
$begingroup$
For $n geq 2$,
$$bigveelimits_{j=2}^n big(bigveelimits_{i=1}^{j-1}(a_i land a_j) big)$$
might satisfy your formula.
answered Jan 31 at 2:55
Doyun NamDoyun Nam
66619
$endgroup$
For $n geq 2$,
$$bigveelimits_{j=2}^n big(bigveelimits_{i=1}^{j-1}(a_i land a_j) big)$$
might satisfy your formula.
answered Jan 31 at 2:55
Doyun NamDoyun Nam
66619
answered Jan 31 at 2:55
Doyun NamDoyun Nam
66619
answered Jan 31 at 2:55
Doyun NamDoyun Nam
66619
answered Jan 31 at 2:55
Doyun NamDoyun Nam
66619
66619
add a comment |
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$begingroup$
For $n=3$, does the last term of your formula mean either $a_2 land a_3$ or $a_2 lor a_3$?
$endgroup$
– Doyun Nam
Jan 31 at 2:38
$begingroup$
it was a typo i fixed it
$endgroup$
– user614735
Jan 31 at 2:38
$begingroup$
This is perhaps not the most elegant, but my first instinct was to use $bigveelimits_{{i,j}inbinom{[n]}{2}}(a_iwedge a_j)$, using the notation that $[n]={1,2,3,dots,n}$ and $binom{A}{k}$ with $A$ a set is the set of subsets of size $k$ of $A$. This doesn't work for $n=1$, but should work for all larger $n$. To be fair, the meaning I use for the notation $binom{A}{k}$ is not widely used outside of smaller circles in combinatorics.
$endgroup$
– JMoravitz
Jan 31 at 2:40