Lecture 1

Question 1

a proposition is a “declarative sentence” that declares a fact that is either TRUE or FALSE .

Question 2

we use letters to denote “propositional variables” or “sentential variables” to represent propositions.

Question 3

………………. are formed from existing propositions using ……………..

Answer 3

• compound propositions
• logical operators.

Question 4

……………………are logical operators that are used to form new propositions from two or more existing propositions.

Answer 4

connectives

Question 5

p : “A student who has taken calculus can take this class”
q :“A student who has taken introductory computer
science can take this class.”

students who have taken both calculus and introductory computer science can take the class, as well as the students who have taken only one of the two subjects.
Hence, this statement can be expressed as ( …………. ), also called ………….., or the ………………of p and q.

Answer 5

• p ∨ q
• the inclusive or
• the disjunction of p and q

Question 6

the connective “or” has two meanings :-

1) the inclusive or :- includes the case that both propositions could be true

2) the exclusive or :- excludes the case that both propositions could be true , i.e.:- true when only one of the propositions are true , not both.

Question 7

p → q, is conditional statement(or…………….) where :-
p is called the ………… (or ………. or ……….) .
q is called the ……….. (or …………..).

-One way to view the logical conditional is to think of an obligation or contract.
“If I am elected, then I will lower taxes.”
If the politician is elected and does not lower taxes, then the politician has broken the campaign pledge.

• the only case in the truth table that gives false is when :- TRUE → FALSE
  p : politician is elected. (TRUE )
  q : lowered taxes. (FALSE)

Answer 7

• implication.
• hypothesis , antecedent, premise
• conclusion , consequence

Question 8

p → q == NOT P OR Q
“q is NECESSARY for p”
“p is sufficient for q “ but NOT NECESSARY

Question 9

the converse of P →q is ………….
and it is a false statement it is NOT equivalent to the original statement.

Answer 9

q →P

Question 10

the contrapositive of P →q is ………..

Only the contrapositive is equivalent to the original statement.

Answer 10

¬q → ¬ P

Question 11

the inverse of P →q is ………………
and it is a false statement it is NOT equivalent to the original statement.

Answer 11

¬ P → ¬ q

Question 12

Find the converse, inverse, and contrapositive of
“It raining is a sufficient condition for my not going to town.”

Answer 12

P → q
p : It is raining
q : not go to town.

converse : q →P if i do not go to town then it is raining .

inverse : ¬ P → ¬ q if it is not raining then i will go to town

contrapositive : ¬q → ¬ P if i go to town then it is not raining

Question 13

Find the converse, inverse, and contrapositive of
“The home team wins whenever it is raining.”

Answer 13

P → q : if it is raining , then the home team wins
p : it is raining
q : The home team wins

converse : q →P If the home team wins, then it is raining.

inverse : ¬ P → ¬ q If it is not raining, then the home team does not win.

contrapositive : ¬q → ¬ P If the home team does not win, then it is not raining.

Question 14

P → q if i live in Cairo then i live in Egypt .
p : i live in Cairo
q : i live in Egypt .

• converse : q →P if i live in Egypt , then i live in Cairo .
• a converse is a reverse of the conditional statement.
• it is a false statement because i could be living in any other governate other than Cairo.

NOTE : if the converse of the implication was TRUE , then it flows both ways and it is a biconditional statement. p ↔ q

Question 15

P → q if i live in Cairo then i live in Egypt .
p : i live in Cairo
q : i live in Egypt

• inverse : ¬ P → ¬ q if i don’t live in Cairo then i don’t live in Egypt .
• inverse is just a negation of the sentence.
• it is a false statement because i could be living in Egypt in any other governate other than Cairo .

NOTE : the inverse and the converse have the same truth value , if one is false , so is the other.

Question 16

P → q if i live in Cairo then i live in Egypt .
p : i live in Cairo
q : i live in Egypt

• contrapositive : ¬q → ¬ P if i don’t live in Egypt ,then i don’t live in Cairo.
• contrapositive is a reverse negation of the conditional statement .
• it is a TRUE statement , because the conditional statement is also true , they both have the same truth value.

Question 17

The biconditional statement p ↔ q is the
proposition “……………” or
“……………………..” or
“……………………….”

The biconditional statement p ↔ q is true
when p and q have ……………, and is false otherwise.

Biconditional statements are also called
…………….

Answer 17

• ” p if and only if q “
  “ p is necessary and sufficient for q “
  “ if p then q , and conversely “
• p and q have the same truth values
• bi-implications.

Question 18

Two propositions are ……….. if they ALWAYS have the same truth value.

the conditional is equivalent to the ……….

Answer 18

equivalent
contrapositive

Question 19

the rows in the truth table represents all the possible combinations of values for the atomic propositions.

this means that :-
with n atomic propositions,
we can construct ….. distinct (i.e., not equivalent) propositions.

Answer 19

2^n

Question 20

Precedence of Logical Operators:-
1- NOT
2- AND
3- OR
4- IMPLICATION
5- BICONDITIONAL

Question 21

A variable is called a ……….. if its value is either true or false & can be represented using a ……..

Answer 21

• Boolean variable , bit

Question 22

System specifications should be ……….., that is, they should not contain conflicting requirements that could be used to derive a contradiction.

Answer 22

consistent

Question 23

Translating English Sentences :-

“If I go to Harry’s or to the country, I will not go shopping.”

p: I go to Harry’s.
q: I go to the country.
r: I will go shopping.

Answer 23

(p ∨ q) → ¬ r

Question 24

Translating English Sentences :-

“You cannot ride the roller coaster if you are under 4 feet tall unless you are older than 16 years old.”

q: “You can ride the roller coaster”
r: “You are under 4 feet tall”
s: “You are older than 16 years old”

Answer 24

r ^ ¬s → ¬q

Question 25

Translating English Sentences :-

“You can access the Internet from campus only if you are a computer science major or you are not a freshman.”

a: “You can access internet from campus”
c: “You are a computer science major”
f: “You are a freshman.”

Answer 25

a → (c ∨ ¬ f )

Question 26

Translating English Sentences :-

p: “The message is scanned for viruses” and q: “The message was sent from an unknown system”

a) “The message is scanned for viruses whenever the message was sent from an unknown system.”

Answer 26

Since “whenever” means “if”
q → p

Question 27

Translating English Sentences :-

p: “The message is scanned for viruses” and q: “The message was sent from an unknown system”

b) “The message was sent from an unknown system but it was not scanned for viruses.”

Answer 27

Since “but” means “and”
q ^ ¬ p

Question 28

Translating English Sentences :-

p: “The message is scanned for viruses” and q: “The message was sent from an unknown system”

c) “It is necessary to scan the message for viruses whenever it was sent from an unknown system.”

Answer 28

q → p

Question 29

Translating English Sentences :-

p: “The message is scanned for viruses” and q: “The message was sent from an unknown system”

d) “When a message is not sent from an unknown system it is not scanned for viruses.”

Answer 29

“when” means “if”
¬q → ¬p