To determine whether $R$ is left-total and functional, we will analyze the properties of the relation $R$ defined between sets $A = \{2, 4, 6\}$ and $B = \{\heartsuit, \diamondsuit\}$. Let's recall what these properties mean:

1. Left-total: For every element $a \in A$, there is at least one $b \in B$ such that $(a, b) \in R$.

2. Functional: For every $a \in A$, there is at most one $b \in B$ such that $(a, b) \in R$.

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Step 1: Analyze the Relation $R$

Suppose $R$ is explicitly defined, or we infer the properties from how $R$ is described. Without specific pairs in $R$, we assume it adheres to the properties based on a theoretical justification.

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Checking Left-Total Property

• For $R$ to be left-total, every element of $A = \{2, 4, 6\}$ must be related to at least one element in $B = \{\heartsuit, \diamondsuit\}$.
• This means:
  • $2$ must relate to some element in $B$ (e.g., $(2, \heartsuit)$).
  • $4$ must relate to some element in $B$ (e.g., $(4, \diamondsuit)$).
  • $6$ must relate to some element in $B$ (e.g., $(6, \heartsuit)$).

If all elements in $A$ have at least one pair in $R$, then $R$ is left-total.

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Checking Functional Property

• For $R$ to be functional, each element of $A$ must relate to exactly one element in $B$, or at least at most one.
• This means:
  • For $2$, there should be exactly one pair $(2, b) \in R$ with $b \in B$.
  • For $4$, there should be exactly one pair $(4, b) \in R$.
  • For $6$, there should be exactly one pair $(6, b) \in R$.

If no element in $A$ is related to more than one element in $B$, then $R$ is functional.

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Step 2: Conclusion

If $R$ satisfies both the above conditions, then it is left-total and functional.

Tip: To verify this fully, we need the explicit definition of $R$ (e.g., specific pairs). Double-check the exact set $R$.

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Would you like me to generate an example relation $R$ to confirm this analysis, or explain further details?

Expanding Questions:

1. How is the relationship $R$ defined explicitly (e.g., list of pairs)?
2. Can we represent $R$ as a table or diagram for clarity?
3. Are there examples of relations that are left-total but not functional?
4. What is the difference between functional and injective relations?
5. How do these properties change when $A$ and $B$ have more elements?

Tip: Always represent relations graphically or as a set of ordered pairs to easily verify these properties.