Homework #2
Exercises
1. Mark each statement True or False. Justify each answer.
(a) The symbol ∀ means ”for every.”
(b) The negation of a universal statement is another universal statement.
2. Mark each statement True or False. Justify each answer.
(a) The symbol ”∃” means ”there exist several”
(b) If a variable is used in the antecedent of an implication without being quantified, then
the universal quantifier is assumed to apply.
(c) The order in which quantifiers are used affects the truth value.
3. Write the negation of each statement.
(a) Some pencils are red.
(b) All chairs have four legs.
(c) No one one the basketball team is over 6 feet 4 inches tall.
(d) ∃ x > 2 such that f (x) = 7.
(e) ∀ x ∈ A, ∃ y > 2 such that 0 < f(y) < f(x).
(f) If x > 3, then ∃ E > 0 such that x2 > 9+ E.
4. Write the negation of each statement.
(a) Everyone likes Robert.
(b) Some students work part time.
(c) No square matrices are triangular.
(d) ∃ x in B such that f(x) > k.
(e) If x > 5, then f(x) < 3 or f(x) > 7.
(f) x is in A, then ∃ y in B such that f(x) < f(y).
5. Determine the truth value of each statement, assuming x is a real number. Justify
your answer.
(a) ∃ x in the interval [2, 4] such that x < 7.
(b) ∀ x in the interval [2, 4], x < 7.
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(c) ∃ x ∈ R such that x2 = 5.
(d) ∀ x ∈ R , x2 = 5.
(e) ∃ x ∈ R such that x2 =
/ -3.
(f) ∀ x ∈ R, x2 =
/ -3.
(g) ∃ x ∈ R 3 x ÷ x =1
(h) ∀ x ∈ R, x÷x =1
6. Determine the truth value of each statement, assuming x is a real number. Justify
your answer.
(a) ∃x in the interval [3,5] such that x ≥ 4.
(b) ∀x in the interval [3,5], x ≥ 4.
(c) ∃x ∈ R such that x2 =
/ 3.
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(d) ∀x ∈ R , x =
/ 3.
(e) ∃ x ∈ R such that x2 = -5.
(f) ∀ x ∈ R, x2 = -5.
(g) ∃ x ∈ R such that x-x=0.
(h) ∀ x ∈ R , x-x=0.
7. Below are two strategies for determining the truth value of a statement involving a
positive number x and another statement P(x).
For each statement below, indicate which strategy is appropriate.
(i) Find some x>0 such that P(x) is true.
(ii) Let x be the name for any number greater than 0 and show that P(x) is true.
(a) ∀x > 0, P(x).
(b) ∃ x > 0 such that P(x).
(c) ∃ x > 0 such that ¬ P(x).
(d) ∀x > 0, ¬ P(x).
8. Which of the following best identifies f as a constant function where x and y are real
numbers.
(a) ∃x ∈ R such that ∀y ∈ R f (x) = y
(b) ∀x ∈ R ∃y ∈ R such that f (x) = y
(c) ∃y ∈ R such that ∀x ∈ R, f (x) = y
(d) ∀y ∈ R ∃x ∈ R such that f (x) = y
9. Determine the truth value of each statement, assuming that x and y are real numbers. Justify your answer.
(a) ∀ x,y ∈ R y, x ≤ y.
(b) ∃ x and y ∈ R such that x ≤ y.
(c) ∀ x R, ∃ y R such that x ≤ y.
(d) ∃ x ∈ R such that ∀ y ∈ R x ≤ y.
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10. Determine the truth value of each statement, assuming that x and y are real numbers. Justify your answer.
(a) ∀ x ∈ R, ∃ y ∈ R such that xy = 0
(b) ∀ x ∈ R, ∃ y ∈ R such that xy = 1
(c) ∃ y ∈ R such that ∀ x ∈ R, xy = 1
(d) ∀ x ∈ R, ∃ y ∈ R such that xy = x
11. Determine the truth value of each statement, assuming that x,y, and z are real
numbers. Justify your answer.
(a) ∃ x ∈ R such that ∀ y ∈ R ∃ z ∈ R such that x + y = z
(b) ∃ x and y ∈ R such that ∀ z ∈ R , x + y = z
(c) ∀ x and y ∈ R, ∃ z ∈ R such that y − z = x
(d) ∀ x and y ∈ R, ∃ z ∈ R such that xz=y
(e) ∃ x ∈ R such that ∀ y,z ∈ R, z > y implies that z > x+y.
(f)∀ x ∈ R , ∃ y,z ∈ R such that z > y implies that z > x+y.
12. Determine the truth value of each statement, assuming that x,y, and z are real
numbers. Justify your answer.
(a) ∀ x and y ∈ R, ∃ z ∈ R such that ∈ R x + y = z
(b) ∀ x ∈ R ∃ y ∈ R such that ∀ z ∈ R , x + y = z
(c) ∃ x ∈ R such that ∀ y ∈ R , ∃ z ∈ R such that xz = y
(d) ∀ x and y ∈ R, ∃ z ∈ R such that yz = x
(e) ∀ x ∈ R , ∃ y ∈ R such that ∀ z ∈ R, z > y implies that z (f) ∀ x,y ∈ R, ∃ z ∈ R such
that z > y implies that z > x+y.
For the next few questions
(a) Rewrite the defining conditions in logical symbolism using ∀, ∃ ⇒
(b) Write the negation
13. A function f is even if for every x, f (−x) = f (x)
14. A function f is periodic if there exists a k > 0 such that for every x, f (x + k) = f (x)
15. A function f is increasing if for every x and y, if x ≤ y, then f(x) ≤ f(y)
16. A function f is strictly decreasing if for every x and y, if x< y, then f(x) > f(y)
17. A function f: A → B is injective if for every x and y in A, iff (x) = f (y), then x=y
18. A function f: A → B is surjective if for every y in B there exists an x in A, such that
f (x) = y.
19, A function f: D → R is continuous at c ∈ D if for every E > 0 there is a δ > 0 such
that |f (x) − f (c)| < E whenever |x − c| < δ and x ∈ D.
20. A function f is uniformly continuous at a set S if for every E> 0 there is a δ > 0 such
that |f (x) − f (y)| < E whenever x and y are in S and |x − y| < δ
21. The real number L is the limit of the function f: D → R at the point c if for each E>
3
0 there exists a δ > 0 such that |f (x) − L| < E whenever x ∈ D and 0 < |x − c| < δ
Find the logic connector between the sentences
(a) All men are mortal
(b) All men are immortal
(c) None of the men are mortal
(d) None of the men are immortal
(e) There exist a man that is immortal
(f) There exist a man that is mortal
Let C be the set of cats, c ∈ C
M(c) be the proposition ”c has whiskers”
P(c) be the proposition ”c likes fish”
S(c) be the proposition ”c fears mice”
(1)
(2)
(3)
(4)
(5)
Write the sentence with quantifiers and the set defined
The cats with whiskers always likes fish
It is wrong that the cats which likes fish has whiskers.
None of the cats which like fish fear the mice.
The cats that have whiskers fear the mice.
The cats which fear the mice have no whiskers.
Suppose 1, 2, 3 is true. Make a picture presenting the set of cats. The sentences (1, 2,
3) are true. What can you say about statements 4 and 5?
Let H be the set of men.
We propose the two math translations for the sentence. ”All men are happy and quiet.”
(1) ∀ x ∈ H. (x is happy ∧ x is quiet)
Are these statement the same as the first?
(2) (∀ x ∈ H, x is happy) ∧ (∀ x ∈ H, x is quiet)
Are these statement the same as the first?
(-) Are the 3 propositions the same:
”Happy men are quiet”
(1) ∀ x ∈ H, (x is happy ⇒ x is quiet)
Is these statement the same as the first?
(2) (∀ x ∈ H, x is happy) ⇒ (∀ x ∈ H, x is quiet)
Is these statement the same as the first?
(-) Are the 3 propositions the same?
”There exist a man happy and quiet.”
(1) (∃ x ∈ H, x is happy) ∧ (∃ x ∈ H, x is quiet)
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Is these statement the same as the first?
(2) ∃ x ∈ H,[ (x is happy) ∧ (q is quiet)]
Is these statement the same as the first:
(-) Are the 3 propositions the same?
”All men are not happy”
(1) ∀ x ∈ H, ¬ (x is happy)
Is these statement the same as the first:
(2) ∃ x ∈ H, ¬ (x is not happy)
Is these statement the same as the first:
(3) ¬ (∀ x ∈ H, x is happy)
Is these statement the same as the first:
(-) Are the 3 propositions the same?
Write the negation of the following proposition:
(a) p: All men are kind.
(b) p: All interval of R contains an element of the interval [0,1].
(c)p:∀ p ∈ N ∀ n ∈ Z, p ≥ m
(d) (x2 ≥ 1 ∧ x3 < 1 ∨ (x2 ≤ g ∧ x > 0)
(e) ∀ x ∈ R, [(x < 1) ⇒ (x2 < 1)]
Let E be a set and A, B, subsets in E. The definition of A ⊆ B is ”All elements of A is
an element of B”
Which of these proposition(s) below does it correspond to?
(1) ∀ x ∈ A, x ∈ B
(2) ∀ x ∈ E, (x ∈ A ⇒ x ∈ B)
(3) ∀ y ∈ E, (y ∈ A ⇒ y∈ B)
(4) (∀ x ∈ E, x ∈ A) ⇒ (∀ x ∈ E, x ∈ B)
If f is a function from R to R the definition of ”f is bounded” is ∃ M ∈ R, ∀ x ∈ R,
|f (x)| ≤ M
Give the definition ”f is not bounded”
The definition of ”f is increasing” is (∀ (x, y) ∈ R, x ≥ y ⇒ f(x) ≥ f(y)
Give the definition of ”f is not increasing”
Give the definition of f is decreasing
Let f be a function from {0, 1, 2} to {−1, 3, 5} defined by f(0)=3, f(1)=-1, and f(2)=5
For all these propositions give the negation and say which are true or false. (Justify your
answer)
(a) ∀ i ∈ {0, 1, 2}, f(i) ≥ 0
(b) ∃ i ∈ {0, 1, 2}, f(i) ≥ 0
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(c) ∀ j ∈ {−1, 3, 5}, ∃ i ∈ {0, 1, 2} f(i)=j
(d) ∀ j ∈ {−1, 3}, ∃ i ∈ { 0,1,2 } f(i) =j
(e) ∃ j ∈ {−1, 3, 5} such that ∀ i ∈ {0, 1, 2} f (i) = j.
For all the propositions below. Give the negation and say if the statement is false.
Justify your answer.
(a) ∀ x ∈ R, x2 > 0
(b) ∃ x ∈ R, √
x2 > 0
(c) ∀ x ∈ R, x2 = x
(d) (∀ x ∈ R), (∃ y ∈ R), x + y = 0
(e) (∃ y ∈ R), (∀ x ∈ R), x + y = 0
Write with quantifiers.
(a) All natural numbers have a real square root.
(b) All natural numbers have a positive real number greater than them.
(c) ∃ a real number smaller than all the integers.
(d) The interval I is included in [1,2]
Let F be the of French. We denote, ∀ ∈ F,
P(x), the proposition ”x has brown hats”
Q(x), the proposition ”x is tall”
Answer the following questions
(1) In a picture, represent in F the set of elements of F such that P(x) is true then the set
of elements such that Q is true.
(2) Consider the following sentence:
(∀ x ∈ F) [P-[(x) or Q(x)]
and (∀ x ∈ F, P(x)) or (∀ x ∈ F, Q(x))
Tell if the propositions are true or fall in the case of the picture.
(∀ x ∈ F) [P(x) or Q(x)]
(∀ x ∈ F, P(x)) or (∀ x ∈ F, Q(x))
If you worked to represent when each of the sentence us true how would you represent if
they are the same?
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Let E be a nonempty set. Denote P(x) and Q(x) as 2 propositions. Consider the following propositions.
∃ x ∈ E, P(x) and Q(x)
(∃ x ∈ E, P(x)) and (∃ x ∈ E, Q(x))
Interpret the formulas P(x) and Q(x) to be pawns of the chess game. P(x) is ”x is black”
and Q(x), ”x is white”
Express in english the two sentences
∃ x ∈ E, P(x) and Q(x)
(∃ x ∈ E, P(x)) and (∃ x ∈ E, Q(x))
Tell me if you think the two propositions are the same or not.
Write the negation of the two propositions
∃ x ∈ E, P(x) and Q(x)
(∃ x ∈ E, P(x))and (∃ x ∈ E, Q(x))
Give the logic relation between these two propositions.
∃ x ∈ E, P(x) and Q(x) ⇒ (∃ x ∈ E, P(x)) and (∃ x ∈ E, Q(x))
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