Search & Filtering
Express each of these statements using quantifiers. Then form the negation of the statement, so that no negation is to the left of a quantifier. Next, express the negation in simple English. (Do not simply use the phrase “It is not the case that.”) a) Some old dogs can learn new tricks. b) No rabbit knows calculus. c) Every bird can fly. d) There is no dog that can talk. e) There is no one in this class who knows French and Russian.
Express each of these statements using quantifiers. Then form the negation of the statement so that no negation is to the left of a quantifier. Next, express the negation in simple English. (Do not simply use the phrase “It is not the case that.”) a) All dogs have fleas. b) There is a horse that can add. c) Every koala can climb. d) No monkey can speak French. e) There exists a pig that can swim and catch fish.
Translate each of these statements into logical expressions using predicates, quantifiers, and logical connectives.
a) No one is perfect. b) Not everyone is perfect. c) All your friends are perfect. d) At least one of your friends is perfect. e) Everyone is your friend and is perfect. f ) Not everybody is your friend or someone is not perfect.
Let C(x) be the statement “x has a cat,” let D(x) be the statement “x has a dog,” and let F(x) be the statement “x has a ferret.” Express each of these statements in terms of C(x), D(x), F(x), quantifiers, and logical connectives. Let the domain consist of all students in your class. a) A student in your class has a cat, a dog, and a ferret. b) All students in your class have a cat, a dog, or a ferret. c) Some student in your class has a cat and a ferret, but not a dog. d) No student in your class has a cat, a dog, and a ferret. e) For each of the three animals, cats, dogs, and ferrets, there is a student in your class who has this animal as a pet.
Translate these statements into English, where C(x) is “x is a comedian” and F(x) is “x is funny” and the domain consists of all people. a) ∀x(C(x) → F(x)) b) ∀x(C(x) ∧ F(x)) c) ∃x(C(x) → F(x)) d) ∃x(C(x) ∧ F(x))
Let P(x) be the statement “x spends more than five hours every weekday in class,” where the domain for x consists of all students. Express each of these quantifications in English.
a) ∃xP(x) b) ∀xP(x) c) ∃x ¬P(x) d) ∀x ¬P(x)
Determine whether each of these sets is the power set of a set. where a and b are distinct elements.
a) ∅
b) {∅.{∅}}
c) {∅, {a}. {∅, a)}
d) {∅, {a}. {b}. {a, b}}
Let A and B be two non-empty sets. Show that A × B = B × A if and only if A = B.
1.Specify the set A by listing its elements, where
A = {x: x is a whole number less than 100 and divisible by 16}.
2. Specify the set B in set-builder form by giving a written description of its elements,
where B = {0, 1, 4, 9, 16, 25}.
3. Consider
A = {m, a,t, h} C = {x: x = 3n, 1 ≤ n ≤ 4, n ∈ N}
B = {s,t, e, m} D = {x: x = 2n, 1 ≤ n ≤ 6, n ∈ N}
a. What is A ∩ B? b. What is C ∪ D?
5. Solve the following problem using a Venn diagram: Consider the following data
among 110 students in the college dormitory: 30 students are on a list A (taking
Accounting); 35 students are on a list B (taking Biology); and 20 students are on both
lists. Find the number of students:
a. on list A or B
b. on exactly one of the two lists
c. on neither list
1. Write the following sentences in the form of if... then and then find the inverse, converse,
and contrapositive [8]
i) You can become successful and happy unless you lead a lazy life.
ii) We enter the password or we click on the forget password button is necessary for
us to login into the system and create a user profile.
2. With all the steps find out if the following system specifications are consistent or
not- [7]
i) If you study well, then you will get a good grade.
ii) We did all the assignments is necessary for us to get a good grade.
iii) We didn’t do all the assignments.
3. Use any of the two proof methods to prove the following- [5]
∼((∼a ∧ b) ∨ (∼a ∧ ∼b)) ∨ (a ∧ b) ≡ a
