Mixing
From infinitely many equations to one equation with infinite series
Let $Gamma$, $Delta$ and $X$ be infinite but enumerable sets of variables assigned a value in $[0,1]$, where $gamma_i$, $chi_i$ and $delta_i$ refer to the $i$th variable in each set.
Suppose that for each triple $langlegamma_i,chi_i,delta_irangle$,
$$1-min(1,(1-gamma_i)+(1-chi_i))leqmin(1,delta_i)$$
Does this (infinite) collection of equations imply that
$$1-min(1,sum^infty_{i=0}(1-gamma_i)+(1-min(1,sum^infty_{i=0}(chi_i))))leqmin(1,sum^infty_{i=0}(delta_i))$$
Some background information:
This is actually for a soundness proof in mathematical logic I'm struggling with. The proof concerns a sequent calculus with infinitary rules and models with formulas assigned values in $[0,1]$. Intuitively the above inference seems valid but I'm not capable of proving it. I've tried to approach the problem both directly and contrapositively. In the direct case, I get stuck with an infinite series of $-1$, and in the contrapositive case, it seems that the falsity of each of the first inequalities requires a certain relationship between $gamma_i$, $chi_i$ and $delta_i$ which I'm not able to deduce from the falsity of the second inequality.
sequences-and-series algebra-precalculus
asked Nov 21 '18 at 17:16
Andy Mount
112
add a comment |
Let $Gamma$, $Delta$ and $X$ be infinite but enumerable sets of variables assigned a value in $[0,1]$, where $gamma_i$, $chi_i$ and $delta_i$ refer to the $i$th variable in each set.
Suppose that for each triple $langlegamma_i,chi_i,delta_irangle$,
$$1-min(1,(1-gamma_i)+(1-chi_i))leqmin(1,delta_i)$$
Does this (infinite) collection of equations imply that
$$1-min(1,sum^infty_{i=0}(1-gamma_i)+(1-min(1,sum^infty_{i=0}(chi_i))))leqmin(1,sum^infty_{i=0}(delta_i))$$
Some background information:
This is actually for a soundness proof in mathematical logic I'm struggling with. The proof concerns a sequent calculus with infinitary rules and models with formulas assigned values in $[0,1]$. Intuitively the above inference seems valid but I'm not capable of proving it. I've tried to approach the problem both directly and contrapositively. In the direct case, I get stuck with an infinite series of $-1$, and in the contrapositive case, it seems that the falsity of each of the first inequalities requires a certain relationship between $gamma_i$, $chi_i$ and $delta_i$ which I'm not able to deduce from the falsity of the second inequality.
sequences-and-series algebra-precalculus
asked Nov 21 '18 at 17:16
Andy Mount
112
add a comment |
Let $Gamma$, $Delta$ and $X$ be infinite but enumerable sets of variables assigned a value in $[0,1]$, where $gamma_i$, $chi_i$ and $delta_i$ refer to the $i$th variable in each set.
Suppose that for each triple $langlegamma_i,chi_i,delta_irangle$,
$$1-min(1,(1-gamma_i)+(1-chi_i))leqmin(1,delta_i)$$
Does this (infinite) collection of equations imply that
$$1-min(1,sum^infty_{i=0}(1-gamma_i)+(1-min(1,sum^infty_{i=0}(chi_i))))leqmin(1,sum^infty_{i=0}(delta_i))$$
Some background information:
This is actually for a soundness proof in mathematical logic I'm struggling with. The proof concerns a sequent calculus with infinitary rules and models with formulas assigned values in $[0,1]$. Intuitively the above inference seems valid but I'm not capable of proving it. I've tried to approach the problem both directly and contrapositively. In the direct case, I get stuck with an infinite series of $-1$, and in the contrapositive case, it seems that the falsity of each of the first inequalities requires a certain relationship between $gamma_i$, $chi_i$ and $delta_i$ which I'm not able to deduce from the falsity of the second inequality.
sequences-and-series algebra-precalculus
asked Nov 21 '18 at 17:16
Andy Mount
112
Let $Gamma$, $Delta$ and $X$ be infinite but enumerable sets of variables assigned a value in $[0,1]$, where $gamma_i$, $chi_i$ and $delta_i$ refer to the $i$th variable in each set.
Suppose that for each triple $langlegamma_i,chi_i,delta_irangle$,
$$1-min(1,(1-gamma_i)+(1-chi_i))leqmin(1,delta_i)$$
Does this (infinite) collection of equations imply that
$$1-min(1,sum^infty_{i=0}(1-gamma_i)+(1-min(1,sum^infty_{i=0}(chi_i))))leqmin(1,sum^infty_{i=0}(delta_i))$$
Some background information:
This is actually for a soundness proof in mathematical logic I'm struggling with. The proof concerns a sequent calculus with infinitary rules and models with formulas assigned values in $[0,1]$. Intuitively the above inference seems valid but I'm not capable of proving it. I've tried to approach the problem both directly and contrapositively. In the direct case, I get stuck with an infinite series of $-1$, and in the contrapositive case, it seems that the falsity of each of the first inequalities requires a certain relationship between $gamma_i$, $chi_i$ and $delta_i$ which I'm not able to deduce from the falsity of the second inequality.
sequences-and-series algebra-precalculus
sequences-and-series algebra-precalculus
asked Nov 21 '18 at 17:16
Andy Mount
112
asked Nov 21 '18 at 17:16
Andy Mount
112
asked Nov 21 '18 at 17:16
Andy Mount
112
asked Nov 21 '18 at 17:16
Andy Mount
112
asked Nov 21 '18 at 17:16
Andy Mount
112
112
add a comment |
add a comment |
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