Question

In: Advanced Math

Let t be a positive integer. Prove that, if there exists a Steiner triple system of...

Let t be a positive integer. Prove that, if there exists a Steiner triple system of index 1 having v varieties, then there exists a Steiner triple system having v^t varieties

Solutions

Expert Solution

Let t be a positive integer.


Related Solutions

8.Let a and b be integers and d a positive integer. (a) Prove that if d...
8.Let a and b be integers and d a positive integer. (a) Prove that if d divides a and d divides b, then d divides both a + b and a − b. (b) Is the converse of the above true? If so, prove it. If not, give a specific example of a, b, d showing that the converse is false. 9. Let a, b, c, m, n be integers. Prove that if a divides each of b and c,...
Let n be a positive integer. Prove that if n is composite, then n has a...
Let n be a positive integer. Prove that if n is composite, then n has a prime factor less than or equal to sqrt(n) . (Hint: first show that n has a factor less than or equal to sqrt(n) )
Let F be a finite field. Prove that there exists an integer n≥1, such that n.1_F...
Let F be a finite field. Prove that there exists an integer n≥1, such that n.1_F = 0_F . Show further that the smallest positive integer with this property is a prime number.
6. Let t be a positive integer. Show that 1? + 2? + ⋯ + (?...
6. Let t be a positive integer. Show that 1? + 2? + ⋯ + (? − 1)? + ?? is ?(??+1). 7. Arrange the functions ?10, 10?, ? log ? , (log ?)3, ?5 + ?3 + ?2, and ?! in a list so that each function is big-O of the next function. 8. Give a big-O estimate for the function ?(?)=(?3 +?2log?)(log?+1)+(5log?+10)(?3 +1). For the function g in your estimate f(n) is O(g(n)), use a simple function g...
Let G be an abelian group and n a fixed positive integer. Prove that the following...
Let G be an abelian group and n a fixed positive integer. Prove that the following sets are subgroups of G. (a) P(G, n) = {gn | g ∈ G}. (b) T(G, n) = {g ∈ G | gn = 1}. (c) Compute P(G, 2) and T(G, 2) if G = C8 × C2. (d) Prove that T(G, 2) is not a subgroup of G = Dn for n ≥ 3 (i.e the statement above is false when G is...
7. Let m be a fixed positive integer. (a) Prove that no two among the integers...
7. Let m be a fixed positive integer. (a) Prove that no two among the integers 0, 1, 2, . . . , m − 1 are congruent to each other modulo m. (b) Prove that every integer is congruent modulo m to one of 0, 1, 2, . . . , m − 1.
let d be a positive integer. Prove that Q[sqrt d] = {a + b sqrt d|...
let d be a positive integer. Prove that Q[sqrt d] = {a + b sqrt d| a, b is in Q} is a field. provide explanations.
(3) Let m be a positive integer. (a) Prove that Z/mZ is a commutative ring. (b)...
(3) Let m be a positive integer. (a) Prove that Z/mZ is a commutative ring. (b) Prove that if m is composite, then Z/mZ is not a field. (4) Let m be an odd positive integer. Prove that every integer is congruent modulo m to exactly one element in the set of even integers {0, 2, 4, 6, , . . . , 2m− 2}
Let {an} be a bounded sequence. In this question, you will prove that there exists a...
Let {an} be a bounded sequence. In this question, you will prove that there exists a convergent subsequence. Define a crest of the sequence to be a term am that is greater than all subsequent terms. That is, am > an for all n > m (a) Suppose {an} has infinitely many crests. Prove that the crests form a convergent subsequence. (b) Suppose {an} has only finitely many crests. Let an1 be a term with no subsequent crests. Construct a...
Suppose a is a positive integer and p is a prime/ Prove that p|a if and...
Suppose a is a positive integer and p is a prime/ Prove that p|a if and only if the prime factorization of a contains p. Can someone please show a full proof to this, thank you.
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT