Discrete Mathematics Answers

Suppose S is a set, and P={S} is a partition of S. Find the equivalence relation R corresponding to P.

Let S ={a, b, c, d, e}, and P={{a, b},{c, d},{e}}.
(a) Verify that P really is a partiton of S.
(b) Find the equivalence relation R on S induced by P.

Let S be the set of bit strings of length no larger than 6, and define an equivalence relation R on S as follows: (x, y) ϵ R if and only if x and y are of the same length. Specify the partition P of S that arises from R.

Let R be a symmetric relation on a finite set A, and let MR be the bit matrix representing R. Is MR necessarily a symmetric matrix? Why or why not?

Let R be a reflexive relation on a finite set A, and let MR be the bit matrix representing R. Specify the value of the entries on the main diagonal.

Let S be a set with 6 elements and let a and b be distinct elements of S. How many relations R are there on S such that...
(a) (a, b) ϵ R?
b) (a; b) ∉ S?
(c) no ordered pair in R has a as its first element?
(d) at least one ordered pair in R has a as its first element?
(e) no ordered pair in R has a as its first element or b as its second element?

(a) Find the number of relations on the set S={a, b, c, d, e}?
(b) How many relations are there on the set S={a, b, c, d, e} that contain (a, a) and (b, b)?

Show that the relation R=∅ on the empty set S=∅ is reflexive, symmetric, and transitive.

Show that the relation R=∅ on a nonempty set S is symmetric and transitive, but not reflexive.