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Let X={1,2,3,4}. Determine whether or no...

Let `X={1,2,3,4}`. Determine whether or not each relation is a function from X into X.
`f={(2,3),(1,4),(2,1),(3,2),(4,4)}`

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To determine whether the relation \( f = \{(2,3), (1,4), (2,1), (3,2), (4,4)\} \) is a function from the set \( X = \{1, 2, 3, 4\} \) into itself, we need to check if every element in the domain \( X \) is associated with exactly one element in the codomain \( X \). ### Step-by-Step Solution: 1. **Identify the Domain and Codomain:** - The domain is \( X = \{1, 2, 3, 4\} \). - The codomain is also \( X = \{1, 2, 3, 4\} \). 2. **List the Pairs in the Relation:** - The relation \( f \) consists of the pairs: - \( (2, 3) \) - \( (1, 4) \) - \( (2, 1) \) - \( (3, 2) \) - \( (4, 4) \) 3. **Check Each Element in the Domain:** - For \( 1 \): It appears in the pair \( (1, 4) \). This is fine as it has a unique image \( 4 \). - For \( 2 \): It appears in the pairs \( (2, 3) \) and \( (2, 1) \). Here, \( 2 \) has two different images \( 3 \) and \( 1 \), which violates the definition of a function. - For \( 3 \): It appears in the pair \( (3, 2) \). This is fine as it has a unique image \( 2 \). - For \( 4 \): It appears in the pair \( (4, 4) \). This is fine as it has a unique image \( 4 \). 4. **Conclusion:** - Since the element \( 2 \) does not have a unique image (it maps to both \( 3 \) and \( 1 \)), the relation \( f \) is **not a function** from \( X \) into \( X \). ### Final Answer: The relation \( f \) is not a function from \( X \) into \( X \).
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