What is the relationship between the characters in Shakespeare's play Hamlet

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In Shakespeare’s Hamlet, there are a number of interesting relationships that help drive the plot forward. Central to the play is Hamlet’s family unit. Hamlet’s father, King Hamlet, is dead when the play begins. His mother, Gertrude, has remarried King Hamlet’s brother, and Hamlet’s uncle, Claudius, by the time Hamlet returns to Denmark. Hamlet’s father’s death, and the overhasty remarriage of his mother to his uncle contribute to Hamlet’s melancholy at the start of the play.

We soon meet another family central to the plot. Polonius is Claudius’s advisor. His son Laertes is a foil for Hamlet. Polonius’s daughter, and Laertes’s sister, Ophelia, is romantically involved with Hamlet at the start of the play. Ophelia heeds her father’s advice and breaks off the relationship with Hamlet. Hamlet views this as a betrayal of sorts, and his cruelty towards her, along with his murder of her father, leads to her madness and eventual suicide.

Hamlet’s friends are also important characters in the play. Rosencrantz and Guildenstern are friends from his childhood who betray Hamlet by spying on him for King Claudius. Hamlet grows frustrated with them, calls them “sponges,” and is able to eventually switch his own execution orders so that they are killed instead. Horatio is his closest, dearest friend. Horatio brings Hamlet to his father’s ghost, helps Hamlet devise a plan to test Claudius’s guilt, and in the end, is there to comfort him as he dies and tells his story to Fortinbras (Hamlet’s Norwegian counterpart). The relationships in the play are significant because they mostly reveal, with the exception of Horatio, how Hamlet was betrayed in some form by the people who were supposed to love him most.

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Gertrude is Hamlet's mother, former wife of Old King Hamlet (who is now dead, and a ghost), and current wife of Old King Hamlet's brother, Claudius.  Therefore, Claudius is Hamlet's uncle and now his step-father as well.  The relationship between Claudius and Hamlet is adversarial at best, and once Old King Hamlet's ghost tells Hamlet that Claudius murdered him, it gets even worse.  Further, the relationship between Hamlet and his mother, Gertrude, is strained because he feels that she betrayed his father by marrying his brother, and so soon after his father's death (fewer than two months).

Polonius is Claudius's closest adviser, and he is also the father of Ophelia, Hamlet's former lover/girlfriend, and Laertes, the man with whom Hamlet eventually duels (at the play's end) and whose poison kills both himself and Hamlet.  Hamlet does not care for Polonius at all, especially after Ophelia follows Polonius's advice and begins to severe ties with him.  After Hamlet kills Polonius, thinking he was Claudius, Laertes returns from abroad with plans to avenge his father's death.  This is why he challenges Hamlet and tips his rapier with the poison.

Horatio is basically Hamlet's best friend from school.  He is legitimately loyal to and loving of Hamlet.  Rosencrantz and Guildenstern are also school friends, but, when Claudius recruits them to spy on Hamlet for him, Hamlet figures it out and refuses to trust them. (They didn't really realize that they were being used.) 

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