1. Logic

2. Definition of Logic : Example :
a proposition is a declarative sentence ( that is a sentence that decelerate a fact ) that is either true or false , but not both
Example :
Riyadh is the capital city of KSA (T)
1 + 1 = (F)
2 + 2 = (T)
All these sentences are Propositions .

3. What is the time now ? Read this carefully x + 1 = 2 x + y = 2
Example :
What is the time now ?
Read this carefully
x + 1 = 2
x + y = 2
All these sentences are not Proposition .
* If p is True (-p ) is Negation .
Example :
Find the negation of the proposition ?

4. "Today is Friday.“ "At least 1 0 inches of rain fell today in Makkah”.
Solution :
“Today is not Friday”.
“Less than 10 inches of rain feel today in Makkah”. p -p T F

5. Definition :
Let p and q be propositions. The conjunction of p and q, denoted by p /\ q, is true when both p and q are true and is false otherwise. p q
p /\ q T F

6. Definition :
Let p and q be propositions. The disjunction of p and q, denoted by p V q , is false when both p and q are false and is true otherwise. p q
p V q T F

7. Example :
Find the conjunction of the propositions p and q where p is the proposition "Today is Friday"
and q is the proposition "It is raining today.“ ?
Solution :
"Today is Friday and it is raining today."

8. Example :
Find the disjunction of the propositions p and q where p is the proposition "Today is Friday"
and q is the proposition "It is raining today.“ ?
Solution :
"Today is Friday or it is raining today."

9. The Exclusive :
Let p and q be propositions. The exclusive or of p and q, denoted by p (+) q, is the proposition that is true when exactly one of p and q is true and is false otherwise. p q
p (+) q T F

10. The Conditional Statement :
Let p and q be propositions. The conditional statement p  q is the proposition "if p, then q ." The conditional statement p  q is false when p is true and q is false, and true otherwise. p q
p  q T F

11. The Conditional Statement :
All the following statement are equivalent to the implication “ p  q” :
"if p, then q"
"if p, q "
"p i s sufficient for q"
"q if p"
"q when p"
"a necessary condition for p is q"
"q unless -'p"
"p implies q"
"p only if q"
"a sufficient condition for q is p"
"q whenever p"
"q is necessary for p"
"q follows from p"

12. Example :
Let p be the statement “Amal learns discrete mathematics“ and q the statement “Amal will find a good job.“ Express the statement p  q as a statement in English.
Solution :
1 - "If Amal learns discrete mathematics, then she will find a good job."
2 - “Amal will find a good job when she learns discrete mathematics."

13. The contrapositive : of p  q is the proposition -q  -p

14. is called the converse of p  q
The proposition q  p
is called the converse of p  q p q
P  q
q  p T F

15. The proposition -p  -q is called the inverse of p  q

16. The biconditional statement :
Let p and q be propositions. The biconditional statement p < q is the proposition "p if
and only if q ."
The biconditional statement p < q is true when p and q have the same truth
values, and is false otherwise. p q
P  q
q  p
P <q T F

17. Student :
Arwa Abdullah Al-Dawood