Basics of Relations

In a set of $5$ numbers the sum of two of these numbers is $6$ more than the sum of the remaining three numbers, whereas the sum of these three numbers is two times one of those two numbers. What definitely is one of those two numbers?

$A=\left\{4,1\right\}$, $B=\left\{1,5\right\}$. Verify commutative property of Cartesian product of sets.

$A=\left\{5,1\right\}$, $B=\left\{1,2\right\}$. Verify commutative property of Cartesian product of sets.

$A=\left\{5,6\right\}$, $B=\left\{6,2\right\}$. Verify commutative property of Cartesian product of sets.

$A=\left\{\mathrm{5},\mathrm{4}\right\}$, $B=\left\{\mathrm{4},3\right\}$. Verify commutative property of Cartesian product of sets.

$A=\left\{1,2\right\}$, $B=\left\{2,3\right\}$. Verify commutative property of Cartesian product of sets.

$A=\left\{p,q\right\}$, $B=\left\{q,r\right\}$ and $C=\left\{p,r\right\}$. Verify associative property of Cartesian product of sets.

$A=\left\{4,3\right\}$, $B=\left\{3,2\right\}$ and $C=\left\{4,2\right\}$. Verify associative property of Cartesian product of sets.

$A=\left\{3,9\right\}$, $B=\left\{9,6\right\}$ and $C=\left\{3,6\right\}$. Verify associative property of Cartesian product of sets.

$A=\left\{1,4\right\}$, $B=\left\{4,5\right\}$ and $C=\left\{1,5\right\}$. Verify associative property of Cartesian product of sets.

$A=\left\{1,2\right\}$, $B=\left\{2,3\right\}$ and $C=\left\{1,3\right\}$. Verify associative property of Cartesian product of sets.

Find the order of the following ordered $n-$tuple.

$(0,1,2,3,4,6,7,8,9,10)$

Find the order of the following ordered $n-$tuple.

$\left(1,2,3,4,6,7,8,9,10\right)$

Find the order of the following ordered $n-$tuple.

$\left(1,2,3,4,6,7\right)$

Find the order of the following ordered $n-$tuple.

$\left(1,2,3,4,6,7,8\right)$

Find the order of the following ordered $n-$tuple.

$\left(1,2,3,4,6,7,8,9\right)$

If $A=\left\{p,q\right\}$, $B=\left\{r,s\right\}$ and $C=\left\{q,t\right\}$, then prove that, $A\times (B\cup C)=(A\times B)\cup (A\times C)$.

If $A=\left\{a,b\right\}$, $B=\left\{m,n\right\}$ and $C=\left\{b,y\right\}$, then prove that, $A\times (B\cup C)=(A\times B)\cup (A\times C)$.