6A - Relations
This section requires us to understand how we organise numbers. This was covered in Section 1A.
6A - Content video: Relations
This video covers the theory and several examples relating to mathematical relations, including domains.
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\((a,b)\) is considered to be an ordered pair of the elements \(a\) and \(b\).
- \(a\) is considered to be the first element
- \(b\) is considered to be the second element.
- The domain is the set of all of the first elements
- The range is the set of all of the second elements.
- \(\left \{ (x,y):y=x^2 \right \}\)
- \(\left \{ (x,y):y=2x+4, x\in \left \{ 0,1,2,3,4 \right \} \right \}\)
Defining and representing relations:
As stated previously, a relation can be defined by a rule, for example: \(\left \{ (x,y):y=2x+4, x\in \left \{ 0,1,2,3,4 \right \} \right \}\)
As stated previously, a relation can be defined by a rule, for example: \(\left \{ (x,y):y=2x+4, x\in \left \{ 0,1,2,3,4 \right \} \right \}\)
- The ordered pairs for this relation are: \(\left \{ (0,4),(1,6),(2,8),(3,10),(4,12) \right \}\)
- The domain is the set: \(\left \{ 0,1,2,3,4 \right \}\)
- The range is the set: \(\left \{ 4,6,8,10,12 \right \}\)
Implied domains:
When a rule is given for a relation but no domain is specified, then we understand that the relation exists overs its implied (maximal) domain. Consider the examples below.
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■ The implied domain of a relation is the largest set of numbers for which the rule has meaning. |
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6A - Content video: Types of relations
This video covers the theory relating to the different types of relations that you will encounter in this course.
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One-to-one relations:
One-to-one relations: each \(x\)-value has a unique \(y\)-value. Examples include:
One-to-one relations: each \(x\)-value has a unique \(y\)-value. Examples include:
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Many-to-one relations:
Many-to-one relations: more than one \(x\)-value maps to the same \(y\)-value. Examples include:
Many-to-one relations: more than one \(x\)-value maps to the same \(y\)-value. Examples include:
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One-to-many relations:
One-to-many relations: one \(x\)-value maps to more than one \(y\)-value. Examples include:
One-to-many relations: one \(x\)-value maps to more than one \(y\)-value. Examples include:
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Many-to-many relations:
Many-to-many relations: more than one \(x\)-value maps to more than one \(y\)-value. Examples include:
Many-to-many relations: more than one \(x\)-value maps to more than one \(y\)-value. Examples include:
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6A - Example 1: Finding the implied domain and range for a relation
Determine the implied domain and range for each of the following relations:
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6A - Example 1: Video solution
6A - Example 1: Practice
Question 1:
Determine the maximal domain and range of each of the following relations.
Question 2: Consider the relation defined by the rule \[y=\frac{1}{(x-h)^2}+k\] State the values of \(h\) and \(k\) if the domain is \(R\)\\(\left \{ -1 \right \}\) and the range is \((2,\infty)\). 6A - Example 1: Solutions
Question 1:
Question 2: \(h=-1\) and \(k=2\). |