INHAMISSA SECONDARY SCHOOL

SUBJECT: MATHEMATICS 11th Grade SCIENCE FORM: No. 1

THEMATIC UNIT: I. INTRODUCTION TO MATHEMATICAL LOGIC

1. Distinguish from the following expressions the designations of the propositions and give the logical value to the propositions:

The sun is a star. 15÷3×5

b) Mathematics. f)
−3,3
{ }belongs to Z

c) π is an irrational number. g) √ 7≥7

2 2
The bicycle has two wheels. h) 2 +3 ÷13

The Sun is a star j) Every vertebrate is a mammal

f) All mammals are vertebrates

2. Consider the propositions:

António listens to music.

translate each of the following propositions into symbolic language:

a) António listens to music and João practices swimming.

b) Maria does not study Mathematics or António does not listen to music.

c) If João practices swimming then António listens to music

Maria studies Mathematics while João practices swimming.

Neither Maria studies Mathematics nor António listens to music.

If António listens to music, then Maria does not study Mathematics and João practices swimming.

If João does not swim, then Maria studies Mathematics or António listens to music.

h) António listens to music or Maria studies Mathematics but not simultaneously.

Maria studies Mathematics if João does not practice swimming.

It is not true that António listens to music if João swims.

If João does not practice swimming and António does not listen to music, then Maria studies Mathematics.

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MATHEMATICS ES_Inhamissa 2018
INHAMISSA SECONDARY SCHOOL
DISCIPLINE: MATHEMATICS 11th Grade SCIENCE SHEET: № 1
THEMATIC UNIT: I. INTRODUCTION TO MATHEMATICAL LOGIC

3. Consider the propositions:

Maputo is a city.
Paulino wrote a book.
The translation in everyday language of the proposition a ⇒ ~b é :
A. Either Maputo is a city, or Paulino wrote a book.
B. If Maputo is a city, then Paulino wrote a book.
C. If Maputo is a city, then Paulino did not write a book.
D. Maputo is a city and Paulino wrote a book.

I am going to the field.

Translate into everyday language:

a ) p∧~q b)∨ c) p⇒q d )~p⇒ q e)( p∨~p)⇒ q

f)~( p ⇔ q) g )~( p∧q ) h) p¿q i) p ⇒~q j) p∨~q

5. Being p ⇒ q a false proposition, state the logical value of each of the following propositions:

a )~p∧qb)~ ( p∨q ) c )~p⇒ qd)~p⇔ ( ~p∨q ) e )~ ( p∧q ) ⇔ qf) p⇒ ( ∨p )

5. The logical operation that associates two false propositions into a new true proposition is called...
A. Conjunction B. Inclusive disjunction C. Equivalence D. Denial
6. What is the negation of the proposition I s 3 g r e a t e r t h a n o r e q u a l t o 4 ?
A. 3> 4 B. 3 i s l e s s t h a n oC.r e3q iusa ln ot ot e4q u a l t D.
o 43. i s g r e a t e r t h a n o r e q u a l t o 4
7. All even numbers are prime. Its negation is:
A. Any even number is prime B. There is at least one even prime number
Not all even numbers are prime. D. At least one even number is prime
8. x i mitpisl ifalse
e s yif...
A. xey false forums B. x for false y and true
C. xey false forms D. x for true and false y
9. The expressionp ∧ ( q or not, having p
p) false, is identical to...
A. F B. V C. p ∨ F D. p or not F
10. Let t and s be any two propositions. What is the equivalent expression to ∼ ( t⇒s )
A. t or s¬ t ∧ s t ∧ ¬ B. s t or s C. D.
?
11. If p and q are two true propositions then...

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MATHEMATICS ES_Inhamissa 2018
INHAMISSA SECONDARY SCHOOL
DISCIPLINE: MATHEMATICS 11th Grade SCIENCE CARD: No. 1
THEMATIC UNIT: I. INTRODUCTION TO MATHEMATICAL LOGIC

A. p o r it's
q false B. ∼( p a n dit qis) true I ffalse
C.not p or not q it's false D. it's p, then q
12. what is the negation of the proposition p a n d q ?
A. ∼p ∨ q p ∨ ∼q B. C. p and not q D.not p and not q
14. Consider the propositions:
Samora Machel was the 1st President of independent Mozambique
Mozambique is an African country
What is the symbolic writing of:
Samora Machel was the 1st President of independent Mozambique and Mozambique is not an African country.
A. p a n d q B. not p and q C. p and not q D.∼( p and q)
15. Consider the following propositions
15 is a prime number 15 is an odd number
Which of the following statements has a true logical value?
A. ~ p B. p a n d q C. p o r q D. p i f and only if q
16. Which of the propositions is equivalent to p a n d ( p and not q)
A. p AND NOT q, NOT B. p AND q C. p o r q D. p or not q
18. Knowing that p i f aItnisda ofalse
n l yproposition,
if q which of the propositions is false?
A. p a n d q B. p o r q C. ~p ⇒ q D. I f p then q
19. Which of the expressions is equivalent top o r ( not p and q?)
A. ~p ∧ qp ∨ q B. C. ~p AND ~q D.∼p∨∼q
20. Consider the following table: What are the values of x, t, and z, respectively....
P q p and not q p i f a n d o n l y i f q
p q

V V F F F T

V F F V x V

F V V F F Z
A. VVV B. VVF C. FVF
F F V V F F
D. VFV

21. Which of the expressions represents a proposition

A. 2+2 6 B. 2x−1=05+5=15 C. D. x <0
31
I: 4+ 5+3>3;√ III :2x+1=0; IV: 6≤10
22. Consider the following expressions: 5
Which do NOT represent propositions?
A I e II B I e III C II e IV D III e IV
The logical operation that associatestwo false propositions into a new true proposition is called...

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MATHEMATICS ES_Inhamissa 2018
INHAMISSA SECONDARY SCHOOL
DISCIPLINE: MATHEMATICS 11th Grade SCIENCE FORM: No. 1
THEMATIC UNIT: I. INTRODUCTION TO MATHEMATICAL LOGIC

The Conjunction Inclusive disjunction

Implication Equivalence
24. What is the negation of the proposition 3<4 ?

A 3>4 B 3 i s l eCs 3s ti hs anno to re qeDuqa3ulailtsotgo4r e4a t e r t h a n o r e q u a l t o 4

a ⇒ ~b
25The negation of the proposition: é:
A ~ a ∨b B a∧b C a∧~b D ~a∧~b

~(a∨b ) ?
26What is the equivalent proposition?

A a∨b B ~(a∧b ) C ~ a ∨~ b D ~ a ∧~ b

27. What is the equivalent proposition to p∧( p∧~ q ) ?

~p∧qp∧qp∨B C D p∧~q
A

28.Consider the proposition: (~ a ∧b)∨a an equivalent proposition is:

~ a ∧ba∨b ~a∧~b ~a∨~b

A B C D

29. What is the negation of the proposition ~p∧q

?

~p∧~q~p∨~q p∨~q p∧~q

A B C D

30. The negation of the proposition: (a ⇒~b)∧~c it is the proposition:

~a∧b∧c ~a∨(b∧c) ~(a∧b )∧~ c (a∧b)∨c

A B C D

31. The table refers to material equivalence. Under these conditions, what are the dexey values?
p q p⇔ q A
1 1 x x =1ey =1
1 0 0 B
0 1 0 x =0e y =1
0 0 y C
x =1ey =0
D

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MATHEMATICS ES_Inhamissa 2018
INHAMISSA SECONDARY SCHOOL
DISCIPLINE: MATHEMATICS 11th Grade SCIENCES FILE: No. 1
THEMATIC UNIT: I. INTRODUCTION TO MATHEMATICAL LOGIC

x =0ey =0

QUANTIFIERS, EXISTENTIAL & UNIVERSAL

1. Given the expressions:

a )2x+3 b )x +1>0c)x 2 − y 2=0 d ) x+ y =1e)( x− y )( x + y )=0 f) x + y√

1
g)2 x +1<0 h ) (4x+6 ) i) x +1=0 j)5( x +3 ) k ) x≠1− y l) √x 2 +3>0
2
Indicate:

1.1. the designatory expressions;

1.2. as propositional expressions;

1.3. equivalent conditions.

2. Translate into quantified language:

a) All natural numbers are positive.

b) Some real number is negative.

c) There is a natural number greater than 5 and less than 7.

d) There exists a real number xtal thatx+ 3=−1

Between -5 and 5 there are some integer numbers.

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MATHEMATICS ES_Inhamissa 2018
INHAMISSA SECONDARY SCHOOL
DISCIPLINE: MATHEMATICS 11th Grade SCIENCE SHEET: № 1
THEMATIC UNIT: I. INTRODUCTION TO MATHEMATICAL LOGIC

3. Translate the following expressions into plain language and indicate their logical value:
2
a )∀ x, y ∈ IR: x+ y >0 b )∃ x i s a n e l e m e n t o f t h e cn )a t ∃
u rxa l∈n uI m s . ∀ y ∈x∈IR:x
R :b exr=−xd)∀ I R : x≥0
>y
4. Conveniently quantify each of the following conditions in order to obtain propositions

true a ) x + y =3 b ) x 2 + 3x + 1=0 c )|x|≥0 d ) x >y e ) 2− x√+ 5 < 0

5. Without using the negation symbol, negate each of the following propositions:

a )∀ x ∈ I R : x >5∧x<7b)∀ x ∈ I R : x >3∨x>7c)∃ x ∈ R:|x +1|=2d)∃ x∈IR:x2 +1>3∧x≥1

e )∃ x ∈ I R , ∀ y ∈ IR : x + y=2f)2x−3=0⇒ x is in the rational numbers) x 2 +1>0 ⇒ x=2 hours) x∈NaturalNumbers⇒ x ∈ IR

6. Determine the logical value of:

x+2
a )∀ x ∈ I R : |x+3|>0 b)∀ x ∈ I R : x 2 +x+7>0c)∃ x ∈ I R : 2 x 2 −x=0d)∃ x ∈ I N : >2
x−1
1
√
e )∃ x i s a n e l e m e n t o f Z : 2 √+ >3 √f)∃ x ∈ R : x −1+ x+7=6g)
x
~(∃ x ∈√ R : x 2 +1+ x 2 +3=0)

h )∀ x ∈ R:|x+2|≥3i)∀ x, y ∈ IR: x 2 + y 2 ≥0j)∀ x ∈ R : x 2 +2 x >1k)∃ x ∈ IR: 2x 2 +5<0

7. What is the symbolic writing of 'the square of a real number is not negative'?
A. There exists an2x in R such that x.> 0 B. T h e r e e2x i s t s a n x i n R s u c h t h a t x i s g r e a t e r t h a n o r e q u a l t o 0 .
C. For all x in R: 2x is greater than or equal to 0 D. For all x in R: 2x>0
8. What is the negation There of exists a unique x in the real numbers such that x.+ 1=3 ?
A. There exists a unique B. For all xxinin +1such
R: xR i s C.
n that
oFor
t eallqx.−1
xuinaR:l xti−1
os ≠n3−3
o t eD.
q uThere
a l t o exists
3 a unique x in R such that x.−1=3

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