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Disc-a3

# Disc-a3 - 8 possibilities So there are eight such relations...

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1. Give an example of a relation that is symmetric but not transitive . The smallest is f (1 ; 2) ; (2 ; 1) g . 2. Give an example of a relation that is transitive but not symmetric . The smallest is f (1 ; 2) g . 3. How many relations on the set f 1 ; 2 ; 3 g are both re°exive and symmetric ? How many of these are also antisymmetric ? To be re°exive, the relation must contain (1 ; 1) , (2 ; 2) , and (3 ; 3) . Because it is symmetric, (1 ; 2) is in it if and only if it (2 ; 1) is in it, and similarly for (1 ; 3) and (3 ; 1) , and for (2 ; 3) , and (3 ; 2) . So we just have to decide which of (1 ; 2) , (1 ; 3) , and (2 ; 3) will be in it. That±s three yes²no choices for a total of
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Unformatted text preview: 8 possibilities. So there are eight such relations. If the relation is also antisymmetric, then none of (1 ; 2) , (1 ; 3) , or (2 ; 3) can be in it, so there is only one such relation: the equality relation. 4. For the equivalence relation R = f (1 ; 1) ; (2 ; 2) ; (2 ; 3) ; (3 ; 2) ; (3 ; 3) g on f 1 ; 2 ; 3 g , ±nd the equivalence classes [1] and [3] . The equivalence class [1] is the set of elements of f 1 ; 2 ; 3 g that are related to 1 . So [1] = f 1 g . Similarly, [3] = f 2 ; 3 g ....
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