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Discrete Mathematics
Use Mathematical Induction to show that if MR is the bit matrix representing the relation R, then M^[n]R is the matrix representing R^n. (This was how the question was stated. If you're confused about the terms M^[n]R and R^n, they aren't exponentials, the [n] in the first term is meant to be a superscript and the R a subscript. The n in the second term is a superscript.)
Discrete Mathematics
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?
Discrete Mathematics
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.
Discrete Mathematics
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) (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?
Discrete Mathematics
(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)?
(b) How many relations are there on the set S={a, b, c, d, e} that contain (a, a) and (b, b)?
Discrete Mathematics
Show that the relation R=∅ on the empty set S=∅ is reflexive, symmetric, and transitive.
Discrete Mathematics
Show that the relation R=∅ on a nonempty set S is symmetric and transitive, but not reflexive.
Discrete Mathematics
Let A be a square matrix, and AT denotes the transpose of A. Show that the following hold true.
(a) (AT)T=A
(b) (A+B)T=AT+BT
(c) (AB)T=BT AT.
(a) (AT)T=A
(b) (A+B)T=AT+BT
(c) (AB)T=BT AT.
Discrete Mathematics
Let A=[top(1 1) bottom (0 1)] be a matrix consisting of real numbers (we are not interpreting this as a bit matrix). Find a formula for An, where n ϵ Z+, and use Mathematical Induction to prove that your formula is correct.
Discrete Mathematics
Let us denote Sn = an + bn + cn for arbitrary numbers a, b, c.
It is known that S1 = 8, S2 = 66, S3 = 536 for some values of
a, b, c. What is the smallest possible value of S242 — S41 S43?
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#332078 on Oct 2022
#332078 on Oct 2022