Note: In the end of this post, I added context. This post is part of a research paper. (I got rid of the 1000, but there was a reason I included it in the original code.)

Suppose, the following values are either:

cV1[1, 2] = 1
cV1[2, 1] = 1
cV2[1, 2] = 1/4
cV2[2, 1] = 1/3
c[1, 2] = 1/4
c[2, 1] = 1/3

where:

cV1[x,y]==cV1[y,x] && cV1[y,x]>cV2[r,y,x] && cV2[y,x]==c[y,x] && 
c[y,x]>cV2[x,y] && cV2[x,y]==c[x,y]

or:

cV1[1, 2] = 1
cV1[2, 1 ] = 1
cV2[1, 2] = 1/3
cV2[2, 1] = 1/4
c[1, 2] = 1/3
c[2, 1] = 1/4

where:

cV1[x,y]==cV1[r,y,x] && cV1[y,x]>cV2[x,y] && cV2[x,y]==c[x,y] && 
c[x,y]>cV2[y,x] && cV2[y,x]==c[y,x]

Whenever:

cV1[1, 2] = 1
cV1[2, 1 ] = 1
cV2[1, 2] = 1/3
cV2[2, 1] = 1/4
c[1, 2] = 1/3
c[2, 1] = 1/4

and:

Q1[x_, y_] := 
 Q1[x, y] = 
  cV1[x, y] >= cV1[y, x] && cV1[y, x] >= cV2[y, x] && 
  cV2[y, x] >= c[y, x] && c[y, x] > c[x, y] && 
  c[x, y] >= cV2[x, y]

 Q2[x_, y_] := 
 Q2[x, y] = 
  cV1[y, x] >= cV1[x, y] && cV1[x, y] >= cV2[x, y] && 
  cV2[x, y] >= c[x, y] && c[x, y] >= c[y, x] && 
  c[y, x] >= cV2[y, x]

 Q3[x_, y_] := 
 Q3[x, y] = 
  cV1[x, y] >= cV1[y, x] && cV1[y, x] >= cV2[x, y] &&
  cV2[x, y] >= c[x, y] && c[x, y] >= cV2[y, x] && 
  cV2[y, x] >= c[y, x]

 Q4[x_, y_] := 
 Q4[x, y] = 
  cV1[y, x] >= cV1[x, y] && cV1[x, y] >= cV2[y, x] && 
  cV2[y, x] >= c[y, x] && c[y, x] >= cV2[x, y] && 
  cV2[x, y] >= c[x, y]

we want

{Q1[1,2],Q2[1,2],Q3[1,2],Q4[1,2]}

to be the following:

{True, False, False, False}

and

{Q1[2,1],Q2[2,1],Q3[2,1],Q4[2,1]}

to be the following:

{False, True, False, False}

However, we get

{True, False, False, True}
{False, True, True, False}

Also, when:

cV1[1, 2] = 1
cV1[2, 1 ] = 1
cV2[1, 2] = 1/3
cV2[2, 1] = 1/4
c[1, 2] = 1/3
c[2, 1] = 1/4

we want

{Q1[1,2],Q2[1,2],Q3[1,2],Q4[1,2]}

to be the following:

{False, False, True, False}

and

{Q1[2,1],Q2[2,1],Q3[2,1],Q4[2,1]}

to be the following:

{False, False, False, True}

However, we get

{False, True, True, False}
{True, False, False, True}

Question: How do we add criteria to Q1, Q2, Q3, Q4 to get:

{True, False, False, False}
{False, True, False, False}
{False, False, True, False}
{False, False, False, True}

without "cheating":

1. we can't use cV1[1000, Min[1,2], Max[1,2]]: I cannot use this notation in my math paper
2. we can use the Sign function (e.g., Sign[c[r,x,y]-c[r,y,x]]) but when I applied it to Q1, Q2, Q3, and Q4, I couldn't get what I wanted.

Context:

I added context due to the comments here.

Note the following:

1. We can change the Q however we want, as long as constants cV1[1,2], cV1[2,1], cV2[1,2], cV2[2,1], c[1,2], c[2,1] are the same for a given value.
2. The 1000 is not relevant for this post. I will get rid of it; however, it was part of the original code for a reason:

In my research paper, I wanted to average a function that is non-Lebesgue integrable on any bounded interval. Since the mean of the function is always undefined, I needed a new way of averaging this function.

After redefining everything, I attempted to answer the problem on Theorem 11 of pg. 15 (because the paper is incomplete after pg. 17, it's best to avoid reading).

The constant c originates from Equation 28, pg. 16 of the research paper:

$$\small{c(G_r^{\star},G_v^{\star\star})=\inf\left\{|1-\mathbf{c_1}|:\forall(\epsilon>0)\exists(\mathbf{c_1}>0)\forall(r\in\mathbb{N})\exists(v\in\mathbb{N})\left(\left|\frac{\left|\mathcal{S}^{\prime}(\varepsilon,G_{r}^{\star})\right|}{\left|\mathcal{S}^{\prime}(\varepsilon,G_{v}^{\star\star})\right|}-\mathbf{c_1}\right|<\varepsilon\right)\right\}}$$

where $(G_r^{\star})=\left(\mathrm{graph}(f_r^{\star})\right)_{r\in\mathbb{N}}$ and $(G_v^{\star\star})=\left(\mathrm{graph}(f_v^{\star\star})\right)_{v\in\mathbb{N}}$ (i.e., $(f_r^{\star})_{r\in\mathbb{N}}$ and $(f_v^{\star\star})_{v\in\mathbb{N}}$ are sequences of bounded functions) and $|\cdot|$ is the cardinality, since $\mathcal{S}^{\prime}(\varepsilon,G_r^{\star})$ and $\mathcal{S}^{\prime}(\varepsilon,G_v^{\star\star})$ takes a sample point from each set in a partition of $G_r^{\star}$ and $G_v^{\star\star}$ (for every $r\in\mathbb{N}$ and $v\in\mathbb{N}$) with equal measures.

(We can compute $c(G_r^{\star},G_v^{\star\star})$ by setting $r$ to large constant, and minimizing $\left|\frac{\left|\mathcal{S}^{\prime}(\varepsilon,G_{r}^{\star})\right|}{\left|\mathcal{S}^{\prime}(\varepsilon,G_{v}^{\star\star})\right|}\right|$ w.r.t. $v$)

In Theorem 11 of pg. 15, we want $V(\varepsilon,G_r^{\star},1)=r!+1$ (i.e., this is the value we want $|\mathcal{S}^{\prime}(\varepsilon,G_r^{\star})|$ to be for every $r\in\mathbb{N}$ whenever $G_r^{\star}\subseteq\mathbb{R}^{2}$, even if not possible) and either:

1. $|\mathcal{S}^{\prime}(\varepsilon,G_r^{\star})|$ and $|\mathcal{S}^{\prime}(\varepsilon,G_r^{\star\star})|$ is chosen using the definitions in this paper, such that:

$$V(\varepsilon,G_r^{\star})\ge|\mathcal{S}^{\prime}(\varepsilon,G_r^{\star})|>|\mathcal{S}^{\prime}(\varepsilon,G_r^{\star\star})|$$

1. $|\mathcal{S}^{\prime}(\varepsilon,G_r^{\star})|$ and $|\mathcal{S}^{\prime}(\varepsilon,G_r^{\star\star})|$ is chosen using the definitions in this paper, such that:

$$V(\varepsilon,G_r^{\star})\ge |\mathcal{S}^{\prime}(\varepsilon,G_r^{\star\star})|>|\mathcal{S}^{\prime}(\varepsilon,G_r^{\star})|$$

where $\frac{(V(\varepsilon,G_r^{\star},1)-1)^2}{|\mathcal{S}^{\prime}(\varepsilon,G_r^{\star\star})|-1}=|\mathcal{S}^{\prime}(\varepsilon,G_r^{\star})|-1$

Hence:

1. cV1[1,2] is $\mathcal{c}(V(\varepsilon,G_r^{\star},1),G_v^{\star})$
2. cV1[2,1] is ${c}(G_r^{\star},V(\varepsilon,G_v^{\star},1))$
3. cV2[1,2] is ${c}(V(\varepsilon,G_r^{\star},1),G_v^{\star\star})$
4. cV2[2,1] is ${c}(G_r^{\star\star},V(\varepsilon,G_v^{\star},1))$
5. c[1,2] is ${c}(G_r^{\star},G_v^{\star\star})$
6. c[2,1] is ${c}(G_r^{\star\star},G_v^{\star})$.

and an example, when $V^{\prime}(\varepsilon,G_r^{\star},1)=r!+1$, $|\mathcal{S}^{\prime}(\varepsilon,G_r^{\star})|=r!+1$, and $|\mathcal{S}^{\prime}(\varepsilon,G_r^{\star\star})|=(4r!)/3+1$ is:

cV1[1, 2] = 1
cV1[2, 1] = 1
cV2[1, 2] = 1/4
cV2[2, 1] = 1/3
c[1, 2] = 1/4
c[2, 1] = 1/3

or when $V^{\prime}(\varepsilon,G_r^{\star},1)=r!+1$, $|\mathcal{S}^{\prime}(\varepsilon,G_r^{\star})|=r!+1$, and $|\mathcal{S}^{\prime}(\varepsilon,G_r^{\star\star})|=(3r!)/4+1$ is

cV1[1, 2] = 1
cV1[2, 1] = 1
cV2[1, 2] = 1/3
cV2[2, 1] = 1/4
c[1, 2] = 1/3
c[2, 1] = 1/4