Binary Relation Foundations

No

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No.

No

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No.

No

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No.

Yes

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives Yes.

No

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No.

No

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No.

No

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No.

No

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No.

No

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No.

Yes

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives Yes.

No

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No.

No

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No.

No

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No.

No

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No.

No

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No.

Yes

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives Yes.

No

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No.

No

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No.

No

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No.

No

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No.

No

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No.

Yes

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives Yes.

No

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No.

No

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No.

No

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No.

∅

Collect only first components that actually occur in R. Dom(R)=∅; the declared source A need not equal this projection.

∅

Collect only first components that actually occur in R. Dom(R)=∅; the declared source A need not equal this projection.

{-10}

Collect only first components that actually occur in R. Dom(R)={-10}; the declared source A need not equal this projection.

{-9, -8}

Collect only first components that actually occur in R. Dom(R)={-9, -8}; the declared source A need not equal this projection.

{-8, -7, -6}

Collect only first components that actually occur in R. Dom(R)={-8, -7, -6}; the declared source A need not equal this projection.

{-6, -5}

Collect only first components that actually occur in R. Dom(R)={-6, -5}; the declared source A need not equal this projection.

∅

Collect only first components that actually occur in R. Dom(R)=∅; the declared source A need not equal this projection.

{-5}

Collect only first components that actually occur in R. Dom(R)={-5}; the declared source A need not equal this projection.

{-4}

Collect only first components that actually occur in R. Dom(R)={-4}; the declared source A need not equal this projection.

{-3, -2}

Collect only first components that actually occur in R. Dom(R)={-3, -2}; the declared source A need not equal this projection.

{-2, -1, 0}

Collect only first components that actually occur in R. Dom(R)={-2, -1, 0}; the declared source A need not equal this projection.

{0, 1}

Collect only first components that actually occur in R. Dom(R)={0, 1}; the declared source A need not equal this projection.

∅

Collect only first components that actually occur in R. Dom(R)=∅; the declared source A need not equal this projection.

{1}

Collect only first components that actually occur in R. Dom(R)={1}; the declared source A need not equal this projection.

{2}

Collect only first components that actually occur in R. Dom(R)={2}; the declared source A need not equal this projection.

{3, 4}

Collect only first components that actually occur in R. Dom(R)={3, 4}; the declared source A need not equal this projection.

{4, 5, 6}

Collect only first components that actually occur in R. Dom(R)={4, 5, 6}; the declared source A need not equal this projection.

{6, 7}

Collect only first components that actually occur in R. Dom(R)={6, 7}; the declared source A need not equal this projection.

∅

Collect only first components that actually occur in R. Dom(R)=∅; the declared source A need not equal this projection.

{7}

Collect only first components that actually occur in R. Dom(R)={7}; the declared source A need not equal this projection.

{8}

Collect only first components that actually occur in R. Dom(R)={8}; the declared source A need not equal this projection.

{9, 10}

Collect only first components that actually occur in R. Dom(R)={9, 10}; the declared source A need not equal this projection.

{10, 11, 12}

Collect only first components that actually occur in R. Dom(R)={10, 11, 12}; the declared source A need not equal this projection.

{12, 13}

Collect only first components that actually occur in R. Dom(R)={12, 13}; the declared source A need not equal this projection.

∅

Collect only first components that actually occur in R. Dom(R)=∅; the declared source A need not equal this projection.

∅

Select relation pairs whose first component lies in C and collect their second components. R[C]=∅; a relation need not have a unique output for each input.

∅

Select relation pairs whose first component lies in C and collect their second components. R[C]=∅; a relation need not have a unique output for each input.

{-6, -5}

Select relation pairs whose first component lies in C and collect their second components. R[C]={-6, -5}; a relation need not have a unique output for each input.

{-4}

Select relation pairs whose first component lies in C and collect their second components. R[C]={-4}; a relation need not have a unique output for each input.

∅

Select relation pairs whose first component lies in C and collect their second components. R[C]=∅; a relation need not have a unique output for each input.

{-3, -2}

Select relation pairs whose first component lies in C and collect their second components. R[C]={-3, -2}; a relation need not have a unique output for each input.

∅

Select relation pairs whose first component lies in C and collect their second components. R[C]=∅; a relation need not have a unique output for each input.

∅

Select relation pairs whose first component lies in C and collect their second components. R[C]=∅; a relation need not have a unique output for each input.

∅

Select relation pairs whose first component lies in C and collect their second components. R[C]=∅; a relation need not have a unique output for each input.

{2}

Select relation pairs whose first component lies in C and collect their second components. R[C]={2}; a relation need not have a unique output for each input.

{2}

Select relation pairs whose first component lies in C and collect their second components. R[C]={2}; a relation need not have a unique output for each input.

{3, 4}

Select relation pairs whose first component lies in C and collect their second components. R[C]={3, 4}; a relation need not have a unique output for each input.

∅

Select relation pairs whose first component lies in C and collect their second components. R[C]=∅; a relation need not have a unique output for each input.

{5}

Select relation pairs whose first component lies in C and collect their second components. R[C]={5}; a relation need not have a unique output for each input.

{6, 7}

Select relation pairs whose first component lies in C and collect their second components. R[C]={6, 7}; a relation need not have a unique output for each input.

{8}

Select relation pairs whose first component lies in C and collect their second components. R[C]={8}; a relation need not have a unique output for each input.

∅

Select relation pairs whose first component lies in C and collect their second components. R[C]=∅; a relation need not have a unique output for each input.

{9, 10}

Select relation pairs whose first component lies in C and collect their second components. R[C]={9, 10}; a relation need not have a unique output for each input.

∅

Select relation pairs whose first component lies in C and collect their second components. R[C]=∅; a relation need not have a unique output for each input.

∅

Select relation pairs whose first component lies in C and collect their second components. R[C]=∅; a relation need not have a unique output for each input.

∅

Select relation pairs whose first component lies in C and collect their second components. R[C]=∅; a relation need not have a unique output for each input.

{14}

Select relation pairs whose first component lies in C and collect their second components. R[C]={14}; a relation need not have a unique output for each input.

{14}

Select relation pairs whose first component lies in C and collect their second components. R[C]={14}; a relation need not have a unique output for each input.

{15, 16}

Select relation pairs whose first component lies in C and collect their second components. R[C]={15, 16}; a relation need not have a unique output for each input.

∅

Select relation pairs whose first component lies in C and collect their second components. R[C]=∅; a relation need not have a unique output for each input.

∅

Exchange every (x,y) for (y,x), giving R⁻¹=∅. A converse relation is not automatically an inverse function.

∅

Exchange every (x,y) for (y,x), giving R⁻¹=∅. A converse relation is not automatically an inverse function.

{(-6, -10), (-5, -10)}

Exchange every (x,y) for (y,x), giving R⁻¹={(-6, -10), (-5, -10)}. A converse relation is not automatically an inverse function.

{(-4, -9), (-4, -8)}

Exchange every (x,y) for (y,x), giving R⁻¹={(-4, -9), (-4, -8)}. A converse relation is not automatically an inverse function.

{(-4, -8), (-3, -7), (-2, -6)}

Exchange every (x,y) for (y,x), giving R⁻¹={(-4, -8), (-3, -7), (-2, -6)}. A converse relation is not automatically an inverse function.

{(-3, -6), (-2, -5)}

Exchange every (x,y) for (y,x), giving R⁻¹={(-3, -6), (-2, -5)}. A converse relation is not automatically an inverse function.

∅

Exchange every (x,y) for (y,x), giving R⁻¹=∅. A converse relation is not automatically an inverse function.

{(-1, -5)}

Exchange every (x,y) for (y,x), giving R⁻¹={(-1, -5)}. A converse relation is not automatically an inverse function.

{(0, -4), (1, -4)}

Exchange every (x,y) for (y,x), giving R⁻¹={(0, -4), (1, -4)}. A converse relation is not automatically an inverse function.

{(2, -3), (2, -2)}

Exchange every (x,y) for (y,x), giving R⁻¹={(2, -3), (2, -2)}. A converse relation is not automatically an inverse function.

{(2, -2), (3, -1), (4, 0)}

Exchange every (x,y) for (y,x), giving R⁻¹={(2, -2), (3, -1), (4, 0)}. A converse relation is not automatically an inverse function.

{(3, 0), (4, 1)}

Exchange every (x,y) for (y,x), giving R⁻¹={(3, 0), (4, 1)}. A converse relation is not automatically an inverse function.

∅

Exchange every (x,y) for (y,x), giving R⁻¹=∅. A converse relation is not automatically an inverse function.

{(5, 1)}

Exchange every (x,y) for (y,x), giving R⁻¹={(5, 1)}. A converse relation is not automatically an inverse function.

{(6, 2), (7, 2)}

Exchange every (x,y) for (y,x), giving R⁻¹={(6, 2), (7, 2)}. A converse relation is not automatically an inverse function.

{(8, 3), (8, 4)}

Exchange every (x,y) for (y,x), giving R⁻¹={(8, 3), (8, 4)}. A converse relation is not automatically an inverse function.

{(8, 4), (9, 5), (10, 6)}

Exchange every (x,y) for (y,x), giving R⁻¹={(8, 4), (9, 5), (10, 6)}. A converse relation is not automatically an inverse function.

{(9, 6), (10, 7)}

Exchange every (x,y) for (y,x), giving R⁻¹={(9, 6), (10, 7)}. A converse relation is not automatically an inverse function.

∅

Exchange every (x,y) for (y,x), giving R⁻¹=∅. A converse relation is not automatically an inverse function.

{(11, 7)}

Exchange every (x,y) for (y,x), giving R⁻¹={(11, 7)}. A converse relation is not automatically an inverse function.

{(12, 8), (13, 8)}

Exchange every (x,y) for (y,x), giving R⁻¹={(12, 8), (13, 8)}. A converse relation is not automatically an inverse function.

{(14, 9), (14, 10)}

Exchange every (x,y) for (y,x), giving R⁻¹={(14, 9), (14, 10)}. A converse relation is not automatically an inverse function.

{(14, 10), (15, 11), (16, 12)}

Exchange every (x,y) for (y,x), giving R⁻¹={(14, 10), (15, 11), (16, 12)}. A converse relation is not automatically an inverse function.

{(15, 12), (16, 13)}

Exchange every (x,y) for (y,x), giving R⁻¹={(15, 12), (16, 13)}. A converse relation is not automatically an inverse function.

∅

Exchange every (x,y) for (y,x), giving R⁻¹=∅. A converse relation is not automatically an inverse function.