Prove: A x B = ∅ iff. A = ∅ or B = ∅

Use the definitions:
x ∈ S ∩ T iff (x ∈ S) ∧ (x ∈ T)
x ∈ S' iff ¬(x ∈ S)
S ⊆ T iff (∀x)(x ∈ S → X ∈ T)
1. Prove line-by-line:
r → ¬(p → q)
assuming that ¬r ∨ ¬q and ¬(q ∧ r) ∧ p
2. A, B, C & A ⊆ B are sets.
Prove:
BnCgAnC
10. Prove that a connected surface M is orientable iff there
exists a nonvanishing differential 2-form on M
10. Prove that a connected surface M is orientable iff there exists a nonvanishing differential 2-form on M.
10. Prove that a connected surface M is orientable iff there exists a nonvanishing differential 2-form on M.
Prove, in R, that (in the generalized sense) {an] converges to u iff u is the only subsequence limit of {an)
Assume n is an integer. Prove that n is odd iff 3n2 + 4 is odd. Remember that to prove p iff q, you need to prove (i) p → q, and (ii) q → p. Use the fact that any odd n can be expressed as 2k + 1 and any even n can be expressed as 2k, where k is an integer. No other assumptions should be made.
Let an and br be sequences. Prove that sequences an and by both converge iff both an + br and an - bre converge.
Define a relation < on Z by m <n iff |m| < |n| or (\m| = |n| 1 m <n) (a) Prove that < is a partial order on Z. (b) A partial order R on a set S is called a total order (or linear order) iff (Vx, Y ES)(x + y + ((x, y) E R V (y,x) E R)) Prove that is a total order on Z. (c) List the following elements in <-increasing order. –5, 2,...
4. (5 points) Let A and B ben x n matrices. Prove that if A and B are skew symmetric, then A - B is skew symmetric. Recall C = [cj] is skew symmetric iff Cij =-Cji.
1. Let V be a vector space with bases B and C. Suppose that T:V V is a linear map with matrix representations Ms(T)A and Me(T) B. Prove the following (a) T is one-to-one iff A is one-to-one. (b) λ is an eigenvalue of T iff λ is an eigenvalue of B. Consequently, A and B have the same eigenvalues (c) There exists an invertible matrix V such that A-V-BV
1. Let V be a vector space with bases B...
1. f : riemann integrabel in [0,inf). prove that f is lebesgue integrable iff the improper integral converges absolutly.
Let R be the relation on N defined by xRy iff 2 divides x+y. R is an equivalence relation. You do not have to prove that R is an equivalence relation. True or False: 3 ∈ 4/R.