ordered pair

Examples of ordered pair in the following topics:

• Graphs of Linear Inequalities

  • If it is $>$ or $<$, then use a dotted or dashed line, since ordered pairs found on the line would result in a false statement.
  • Shading indicates all the ordered pairs in the region that satisfy the inequality.  
  • For example, if the ordered pair is in the shaded region, then that ordered pair makes the inequality a true statement.
  • These overlaps of the shaded regions indicate all solutions (ordered pairs) to the system.
  • All possible solutions are shaded, including the ordered pairs on the line, since the inequality is $\leq$ the line is solid.  

• Graphs of Equations as Graphs of Solutions

  • In an equation where $x$ is a real number, the graph is the collection of all ordered pairs with any value of $y$ paired with that real number for $x$.
  • For example, to graph the equation $x-1=0,$ a few of the ordered pairs would include:
  • After substituting the rest of the values, the following ordered pairs are found:
  • After graphing the ordered pairs and connecting the points, we see that the set of (infinite) points follows this pattern:
  • Construct the graph of an equation by finding and plotting ordered-pair solutions

• Piecewise Functions

  • Allowing $y=f(x)$, where $f(x)=|x|$, some ordered pair examples of $(x,|x|)$ are:
  • Some ordered pair examples are:
  • Those points satisfy the first part of the function and create the following ordered pairs:
  • For the middle part (piece), $f(x)=3$ (a constant function) for the domain $1ordered pairs are:
  • For the last part (piece), $f(x)=x$ for the domain $x>2$, a few ordered pairs are:

• Introduction to Systems of Equations

  • In order for a linear system to have a unique solution, there must be at least as many equations as there are variables.
  • The solution to a system of linear equations in two variables is any ordered pair that satisfies each equation independently.
  • In this example, the ordered pair (4, 7) is the solution to the system of linear equations.
  • We can verify the solution by substituting the values into each equation to see if the ordered pair satisfies both equations.
  • An independent system has exactly one solution pair $(x, y)$.

• Bond Order

  • Bond order is the number of chemical bonds between a pair of atoms.
  • Bond order is the number of chemical bonds between a pair of atoms; in diatomic nitrogen (N≡N) for example, the bond order is 3, while in acetylene (H−C≡C−H), the bond order between the two carbon atoms is 3 and the C−H bond order is 1.
  • Bond order indicates the stability of a bond.
  • In a more advanced context, bond order does not need to be an integer.
  • For a bond to be stable, the bond order must be a positive value.

• Wilcoxon t-Test

  • ., it is a paired difference test).
  • Let $N$ be the sample size, the number of pairs.
  • Exclude pairs with $\left|{ x }_{ 2,i }-{ x }_{ 1,i } \right|=0$.
  • Order the remaining pairs from smallest absolute difference to largest absolute difference, $\left| { x }_{ 2,i }-{ x }_{ 1,i } \right|$.
  • Rank the pairs, starting with the smallest as 1.

• Introduction to Lewis Structures for Covalent Molecules

  • In covalent molecules, atoms share pairs of electrons in order to achieve a full valence level.
  • Other elements in the periodic table react to form bonds in which valence electrons are exchanged or shared in order to achieve a valence level which is filled, just like in the noble gases.
  • It therefore has 7 valence electrons and only needs 1 more in order to have an octet.
  • In order to achieve an octet for all three atoms in CO2, two pairs of electrons must be shared between the carbon and each oxygen.
  • These are 'lone pairs' of electrons.

• The Paradox of the Chevalier De Méré

  • Getting at least one double six with 24 throws of a pair of dice?
  • To make up for this, a pair of dice should be rolled six times for every one roll of a single die in order to get the same chance of a pair of sixes.
  • Now, when you throw a pair of dice, from the definition of independent events, there is a $(\frac{1}{6})^2 = \frac{1}{36}$ probability of a pair of 6's appearing.
  • That is the same as saying the probability for a pair of 6's not showing is $\frac{35}{36}$.
  • Therefore, there is a probability of $(\frac{36}{36}) - (\frac{35}{36})^{24}$ of getting at least one pair of 6's with 24 rolls of a pair of dice.

• Paired observations and samples

  • When two sets of observations have this special correspondence, they are said to be paired.
  • Two sets of observations are paired if each observation in one set has a special correspondence or connection with exactly one observation in the other data set.
  • To analyze paired data, it is often useful to look at the difference in outcomes of each pair of observations.
  • It is important that we always subtract using a consistent order; here Amazon prices are always subtracted from UCLA prices.
  • Using differences between paired observations is a common and useful way to analyze paired data.

• Metal Cations that Act as Lewis Acids

  • Transition metals can act as Lewis acids by accepting electron pairs from donor Lewis bases to form complex ions.
  • The modern-day definition of a Lewis acid, as given by IUPAC, is a molecular entity—and corresponding chemical species—that is an electron-pair acceptor and therefore able to react with a Lewis base to form a Lewis adduct; this is accomplished by sharing the electron pair furnished by the Lewis base.
  • One coordination chemistry's applications is using Lewis bases to modify the activity and selectivity of metal catalysts in order to create useful metal-ligand complexes in biochemistry and medicine.
  • All these metals act as Lewis acids, accepting electron pairs from their ligands.