What does an arrow between two sets mean in a relation diagram?
Conditions
- The diagram shows two sets, typically represented as ovals or circles.
- An arrow connects an element from the first set to an element from the second set.
Reasoning, step by step
- Identify the starting element of the arrow in the first set.
- Identify the ending element of the arrow in the second set.
- Translate the arrow into an ordered pair (start, end).
- Conclude that this ordered pair is an element of the relation R.
Example
The video draws a pink arrow from to in the humans-and-pets example. The speaker explains that this means the first human owns the cat, and later writes that the ordered pair (, ) belongs to the ownership relation.
Common misconceptions
- Thinking an arrow represents a function or a mapping rather than a relation membership.
- Assuming that elements without arrows are not part of the sets; they are just not related in this specific relation.
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Related questions
The video tests the less-than relation with two reversed ordered pairs: (2, 5) and (5, 2). It shows that is true, so the pair (2, 5) belongs to the relation.
Conditions: The relation is the less-than symbol (<) on integers.; The pairs tested are (2, 5) and (5, 2).
No, a relation does not have to contain every possible ordered pair. By definition, a relation is a subset of the Cartesian product , meaning it can include only some of the ordered pairs.
Conditions: A and B are sets.; R is a relation between A and B.
Ordered pairs are used because the order of the elements matters in a relation. The first component of the pair is related to the second component, and swapping them can change whether the relationship holds.
Conditions: The relation is between two sets A and B.; The relationship is directional or asymmetric.
In the definition of a relation, capital letters A and B denote the sets themselves, while lowercase letters a and b denote individual elements belonging to those sets. The ordered pair (a, b) represents a specific element of the Cartesian product , where a is an element of A and b is an element of B.
Conditions: The context is the formal definition of a relation R between sets A and B.; The notation (a, b) ∈ is used.
Yes, elements of the sets can be left out of a relation. A relation is a subset of the Cartesian product, so it does not need to include every possible ordered pair.
Conditions: A relation R is defined between sets A and B.; R is a subset of .
Answers are generated from source material and independently checked. Consult the original video or creator if something is unclear.