Question

(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 negation of the statement that does not use the symbol ¬.

(4) Under the interpretation where x and y are in R − {0} and P (x, y) is “xy ≥ 0”, is the original statement in (3) true or is its negation true?

(5) Is the statement ( ∃x(P (x) ∨ Q(x)) ) → ( (∃xP (x)) ∨ (∃xQ(x)) ) valid? If it is, explain why. If it isn’t, give an interpretation under which it is false.

Answer #1

where in ( AUB) if their is a single T either in A or B truth table is T.

(AB) if their is single F in either A or B truth table is F

for ~ A negation A T becomes F and F becomes T

in () for only TF combination becomes F remaining all are T.

2.
a. In what order are the operations in the following propositions
performed? i. P ∨ ¬q ∨ r ∧ ¬p
ii. P ∧ ¬q ∧ r ∧ ¬p iii. p ↔ q
∧ r → s b. Suppose that x is a proposition generated by
p, q, and r that is equivalent to p ∨ ¬q. Write out x
as a function of p, q, and r, and then give the truth table for
x

Are the statement forms P∨((Q∧R)∨ S) and ¬((¬ P)∧(¬(Q∧ R)∧ (¬
S))) logically equivalent? I found that they were not logically
equivalent but wanted to check. Also, does the negation outside the
parenthesis on the second statement form cancel out with the
negation in front of P and in front of (Q∧ R)∧ (¬ S)) ?

1) Show that ¬p → (q → r) and q → (p ∨ r) are logically
equivalent. No truth table and please state what law you're using.
Also, please write neat and clear. Thanks
2) .Show that (p ∨ q) ∧ (¬p ∨ r) → (q ∨ r) is a tautology. No
truth table and please state what law you're using. Also, please
write neat and clear.

10. Comparing Statements - Practice 2
Complete the truth table for the given propositions. Indicate
each proposition's main operator by typing a lowercase x in box
beneath the column in which it appears. On the right side of the
truth table, indicate whether each row lists identical or opposite
truth values for the two statements. Also indicate which, if any,
rows show that the statements are consistent with a lowercase x.
Finally, answer the questions beneath the truth table about...

1. Construct a truth table for: (¬p ∨ (p → ¬q)) → (¬p ∨ ¬q)
2. Give a proof using logical equivalences that
(p → q) ∨ (q →
r) and (p → r)
are not logically equivalent.
3.Show using a truth table that (p →
q) and (¬q →
¬p) are logically equivalent.
4. Use the rules of inference to prove that the premise
p ∧ (p
→ ¬q) implies
the conclusion ¬q. Number each
step and give the...

Let A and B be true, X, Y, and Z false. P and Q have unknown
truth value. Please, determine the truth value of the propositions
in problem 1. Please, show the process of calculation by using the
letter ‘T’ for ‘true,’ ‘F’ for ‘false,’ and ‘?’ for ‘unknown value’
under each letter and operator. Please underline your answer (truth
value under the main operator) and make it into Bold font
1. [ ( Z ⊃ P ) ⊃ P ]...

Consider the following (true) statement:
“All birds have wings but some birds cannot fly.”
Part 1
Write this statement symbolically as a conjunction of two
sub-statements, one of which is a conditional and the other is the
negation of a conditional.
Use three components (p, q, and r)
and explicitly state what these components correspond to in the
original statement.
Hint: Any statement in the form "some X cannot Y"
can be rewritten equivalently as “not all X can Y,”...

DISCRETE MATHS [ BOOLEANS AND LOGIC] Please answer
all
Exercise 1.7.1: Determining whether a quantified statement about
the integers is true.
infoAbout
Predicates P and Q are defined below. The domain of discourse is
the set of all positive integers.
P(x): x is prime
Q(x): x is a perfect square (i.e., x = y2, for some
integer y)
Indicate whether each logical expression is a proposition. If
the expression is a proposition, then give its truth value.
(c)
∀x Q(x)...

Given statement p q, the statement q p is called its converse.
Let A be the statement: If it is Tuesday, then you come to campus.
(1) Write down the converse of A in English. (2) Is the converse of
A true? Explain. (3) Write down the contrapositive of A in
English
Let the following statement be given: p = “You cannot swim” q =
“You are less than 10 years old” r = “You are with your parents”
(1)...

Question 1
Consider the following compound inequality:
-2 < x < 3
a) Explain why this compound inequality is actually a
conjunction. Write this algebraic statement verbally.
b) Is the conjunction TRUE or FALSE when x = 3? Justify your
answer logically (not algebraically!) by arguing in terms of truth
values.
Question 2
Write verbally the negation of the following statement:
"Red mangos are always sweet while green mangos are sometimes
sour."
Do not forget to apply De Morgan's laws...

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 33 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 2 hours ago

asked 2 hours ago

asked 2 hours ago

asked 2 hours ago

asked 2 hours ago

asked 3 hours ago

asked 3 hours ago

asked 3 hours ago