Question

Prove the statements (a) and (b) using a set element proof and using only the definitions of the set operations (set equality, subset, intersection, union, complement):

(a) Suppose that A ⊆ B. Then for every set C, C\B ⊆ C\A.

(b) For all sets A and B, it holds that A′ ∩(A∪B) = A′ ∩B.

(c) Now prove the statement from part (b)

Answer #1

1) Given set A and {$} where {$} represents set with only one
element. Prove there is bijection between A x {$} and A.
2) Given sets A, B. Prove A x B is equivalent to B x A using
bijection.
3) Given sets A, B, C. Prove (A x B) x C is equvilaent to A x (B
x C) using a bijection.

Using field axioms and order axioms prove the following
theorems
(i) The sets R (real numbers), P (positive numbers) and [1,
infinity) are all inductive
(ii) N (set of natural numbers) is inductive. In particular, 1
is a natural number
(iii) If n is a natural number, then n >= 1
(iv) (The induction principle). If M is a subset of N (set of
natural numbers) then M = N
The following definitions are given:
A subset S of R...

Prove that (A-B) ∪ (B-A) = (A∪B) - (A∩B) using propositional
logic and definitions of set operators. Please state justification
for each step!

Prove Corollary 4.22: A set of real numbers E is closed and
bounded if and only if every infinite subset of E has a point of
accumulation that belongs to E.
Use Theorem 4.21: [Bolzano-Weierstrass Property] A set of real
numbers is closed and bounded if and only if every sequence of
points chosen from the set has a subsequence that converges to a
point that belongs to E.
Must use Theorem 4.21 to prove Corollary 4.22 and there should...

For each of the following statements: if the statement is true,
then give a proof; if the
statement is false, then write out the negation and prove that.
For all sets A;B and C, if B n A = C n A, then B = C.

Using field and order axioms prove the following theorems:
(i) 0 is neither in P nor in - P
(ii) -(-A) = A (where A is a set, as defined in the axioms.
(iii) Suppose a and b are elements of R. Then a<=b if and
only if a<b or a=b
(iv) Let x and y be elements of R. Then either x <= y or y
<= x (or both).
The order axioms given are :
-A = (x...

Prove the following statements using either direct or
contrapositive proof.
18. If a,b∈Z,then (a+b)^3 ≡ a^3+b^3 (mod 3).

(10) Consider the following property: For all sets A, B and C,
(A-B)∩(A-C)=A-(B∪C)
a. Construct a proof of this property using set definitions.
b. Prove this property using a set-membership table, clearly
stating how the table proves the property.
c. Illustrate this property using Venn diagrams, clearly stating
how the diagram proves the property. You must use a separate Venn
diagram for the set on the left hand side of the equal sign, and
for the set on the right...

For each of the following statements, say whether the statement
is true or false.
(a) If S⊆T are sets of vectors, then span(S)⊆span(T).
(b) If S⊆T are sets of vectors, and S is linearly independent,
then so is T.
(c) Every set of vectors is a subset of a basis.
(d) If S is a linearly independent set of vectors, and u is a
vector not in the span of S, then S∪{u} is linearly
independent.
(e) In a finite-dimensional...

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 <...

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