(1) Determine whether the propositions p → (q ∨ ¬r) and (p ∧ ¬q)
→ ¬r...
(1) Determine whether the propositions p → (q ∨ ¬r) and (p ∧ ¬q)
→ ¬r are logically equivalent using either a truth table or laws of
logic.
(2) Let A, B and C be sets. If a is the proposition “x ∈ A”, b
is the proposition “x ∈ B” and
c is the proposition “x ∈ C”, write down a proposition involving a,
b and c that is logically equivalentto“x∈A∪(B−C)”.
(3) Consider the statement ∀x∃y¬P(x,y). Write down a...
Consider p(x) and q(x), where x ∈ U = {1, 2}. If the following
is true,...
Consider p(x) and q(x), where x ∈ U = {1, 2}. If the following
is true, give a rigorous argument. If it is false, give a
counterexample. (Note that “p implies q” is the same as “if p, then
q” and also as “p → q.”) (i) (∀x ∈ U, p(x) → q(x)) implies [ (∀x ∈
U, p(x)) → (∀x ∈ U, q(x)) ] ? What about its converse ? (ii) (∃x ∈
U, p(x) → q(x)) implies [...
1. For each statement that is true, give a proof and for each
false statement, give...
1. For each statement that is true, give a proof and for each
false statement, give a counterexample
(a) For all natural numbers n, n2
+n + 17 is prime.
(b) p Þ q and ~ p Þ ~ q are NOT logically
equivalent.
(c) For every real number x
³ 1, x2£
x3.
(d) No rational number x satisfies
x^4+ 1/x
-(x+1)^(1/2)=0.
(e) There do not exist irrational numbers
x and y such that...
Identify whether the following are true or false. (a) 5 ∈ {1, 2,
{3, 4}, {1},...
Identify whether the following are true or false. (a) 5 ∈ {1, 2,
{3, 4}, {1}, {5}} (b) {5} ∈ {1, {2}, {3, 4}, 5, {5}} (c) {5} ⊆ {1,
{2}, {3, 4}, 5, {5}} (d) {3, 4} ⊆ {1, 2, {3, 4}, {1}, {5}} (e) {1,
2} ⊆ {1, 2, {3, 4}, {1}, {5}} (f) {5} ∈ P(N) (g) {5} ⊆ P(N) (h)
{{5}} ∈ P(N) (i) ∅ ⊆ P(R) (j) ∅ ∈ P(R)
Please answer True or False for the following statements:
1.) In hypothesis testing, if the null...
Please answer True or False for the following statements:
1.) In hypothesis testing, if the null hypothesis is rejected,
then the alternative or motivated hypothesis must also be
rejected.
2.) Under certain conditions, binomial distributions can be
approximated by normal distributions.
3.) Using a normal distribution to approximate binomial
probabilities is extremely accurate, but takes an excessive amount
of time and should be avoided.
4.) A bell-shaped curve is still normal, even if the mean,
median, and mode are not...