Discrete Mathematics
w350500Note that "iff" stands for "if and only if". Also, to show "whether or not" means to give a proof for it if it is true and to give a counter example if it is false. Finally a R b means "a is related to b", i.e. (a,b)∈R.
- Let R be the relation on socks in your drawer defined by the following property:
a R b iff a and b have the same number of stripes- Show whether or not R is reflexive.
- Show whether or not R is symmetric.
- Show whether or not R is transitive.
- Is R an equivalent relation? If it is, how may equivalence classes does it have?
- Let R be the relation on natural numbers defined by the following property:
a R b iff a+b is even- Show whether or not R is reflexive.
- Show whether or not R is symmetric.
- Show whether or not R is transitive.
- Is R an equivalent relation? If it is, how many equivalence classes does it have?
- Let R be the relation on natural numbers defined by the following property:
a R b iff f(a)=f(b), where f(n) is the number of 1's in the binary representation of n- Show whether or not R is reflexive.
- Show whether or not R is symmetric.
- Show whether or not R is transitive.
- Is R an equivalent relation? If it is, how many equivalence classes does it have?
- Let R be the relation on functions that map natural numbers to natural numbers defined by the following property:
fRg iff for all natural numbers n, f(n)≤g(n)- Show whether or not R is reflexive.
- Show whether or not R is symmetric.
- Show whether or not R is transitive.
- Is R an equivalent relation? If it is, how many equivalence classes does it have?
- Let R be the relation on natural numbers defined by the following property:
a R b iff a%7=b%7 (where % is the C++ remainder operation)- Show whether or not R is reflexive.
- Show whether or not R is symmetric.
- Show whether or not R is transitive.
- Is R an equivalent relation? If it is, how many equivalence classes does it have?
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