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0E: Exercises

  • Page ID
    131044
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    Exercise \(\PageIndex{1}\)

     

    Define \(h : \mathbb{Z} \rightarrow \mathbb{Z} \) by \(h(x) = x^2+4 \).  Determine (with reasons) whether or not \(h \) is one−to−one and whether or not \(h \) is onto.  

    Exercise \(\PageIndex{2}\)

    Suppose \(f : A \rightarrow B \) and \(g : B \rightarrow C \) are functions.  Prove or disprove the following statements: 

    1. If \(g \circ f \) is one-to-one then \(g \) is one-to-one. 

    2. If \(g \circ f \) is one-to-one and \(f \) is onto, then \(g \) is one-to-one.

    3.  If \(g \circ f \) is onto and  \(g \)is one-to-one, then  \(f \) is onto.

    Exercise \(\PageIndex{3}\)

    Determine whether or not each of the following binary relations \(R \) on the given set \(A \) is reflexive, symmetric, antisymmetric, or transitive.  If a relation has a certain property, prove this is so; otherwise, provide a counterexample to show that it does not.  If \(R \) is an equivalence relation, describe the equivalence classes of \(A \). 

    1. Let \(S = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\} \). Define a relation \(R \) on \(A = S \times S \) by \((a, b) R (c, d) \) if and only if \(10a + b \le 10c + d \). 

    2. Let \(A = \mathbb{Z} \backslash \{0\} \). Define a relation \(R \) on \(A \), by \(a R b \) if and only if \( ab > 0 \).

    3. Define a relation \(R \) on \(A = \mathbb{Z} \) by \(a R b \) if and only if \(4 | (3a + b) \).

    4. Define a relation \(R \) on \(A = \mathbb{Z} \) by \(a R b \) if and only if \(3 | (a^2 - b^2 ) \). 

    5. Let \(A = \mathbb{R} \), If \(a,b \in \mathbb{R} \), define \(a R b \) if and only if \(a - b \in \mathbb{Z} \). 

    6. Define a relation \(R \) on the set \(\mathbb{Z} \times \mathbb{Z} \) by \((a, b) R (c, d) \) if and only if \(ac = bd \). 

    7. Define a relation \(R \) on \(\mathbb{Z} \) by \(a R b \) if and only if \(2 \mid a^2+b\). .

    8. Define a relation \(R \) on \(\mathbb{Z} \) by \(a R b \) if and only if \(5 | (2a + 3b) \).

    9. Let \(A = \mathbb{R} \), If \(a,b \in \mathbb{R} \), define \(a R b \) if and only if \(a - b \in \mathbb{Q} \). 

     


    This page titled 0E: Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Pamini Thangarajah.

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