Question

Prove this statement or show why it's false (provide a counter example)

∀x(R(x) ∨ S(x)) → (∃xR(x) ∨ ∃yS(y))

Answer #1

Prove the statement true or use a counter-example to explain why
it is false.
Let a, b, and c be natural numbers. If (a*c) does not divide
(b*c), then a does not divide b.

Prove or give a counter-example:
(a) if R ⊂ S and T ⊂ U then T\ S ⊂ U \R.
(b) if R∪S⊂T∪U, R∩S= Ø and T⊂ R, then S ⊂ U.
(c) if R ∩ S⊂T ∩ S then R⊂T.
(d) R\ (S\T)=(R\S) \ T

Prove or give a counter example: If f is continuous on R and
differentiable on R ∖ { 0 } with lim x → 0 f ′ ( x ) = L , then f
is differentiable on R .

In each case below show that the statement is True or give an
example showing that it is False.
(i) If {X, Y } is independent in R n, then {X, Y, X + Y } is
independent.
(ii) If {X, Y, Z} is independent in R n, then {Y, Z} is
independent.
(iii) If {Y, Z} is dependent in R n, then {X, Y, Z} is
dependent.
(iv) If A is a 5 × 8 matrix with rank A...

Prove the following: Theorem. Let R ⊆ X × Y and S ⊆ Y × Z be
relations. Then
1. Range(S ◦ R) ⊆ Range(S), and
2. if Domain(S) ⊆ Range(R), then Range(S ◦ R) = Range(S)

Prove or disprove the following statements. Remember to disprove
a statement you have to show that the statement is false.
Equivalently, you can prove that the negation of the statement is
true. Clearly state it, if a statement is True or False. In your
proof, you can use ”obvious facts” and simple theorems that we have
proved previously in lecture.
(a) For all real numbers x and y, “if x and y are irrational,
then x+y is irrational”.
(b) For...

Mark the following as true or false, as the case may be. If a
statement is true, then prove it. If a statement is false, then
provide a counter-example.
a) A set containing a single vector is linearly independent
b) The set of vectors {v, kv} is linearly dependent for every
scalar k
c) every linearly dependent set contains the zero vector
d) The functions f1 and f2 are linearly
dependent is there is a real number x, so that...

Prove the following statement: If x ∈ R, then x2 + 1
> x.

1. Prove p∧q=q∧p
2. Prove[((∀x)P(x))∧((∀x)Q(x))]→[(∀x)(P(x)∧Q(x))]. Remember to
be strict in your treatment of quantifiers
.3. Prove R∪(S∩T) = (R∪S)∩(R∪T).
4.Consider the relation R={(x,y)∈R×R||x−y|≤1} on Z. Show that
this relation is reflexive and symmetric but not transitive.

For Problems #5 – #9, you willl either be asked to prove a
statement or disprove a statement, or decide if a statement is true
or false, then prove or disprove the statement. Prove statements
using only the definitions. DO NOT use any set identities or any
prior results whatsoever. Disprove false statements by giving
counterexample and explaining precisely why your counterexample
disproves the claim.
*********************************************************************************************************
(5) (12pts) Consider the < relation defined on R as usual, where
x <...

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 26 minutes ago

asked 29 minutes ago

asked 46 minutes ago

asked 47 minutes ago

asked 47 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago