NAME:
YEAR & PROGRAM:
_______________________
DATE: ______________
SCORE: /80
PRE-FINAL EXAMINATION
DISCRETE MATHEMATICS
3BSCPE & 1IT
PART 1: CONSTRUCT A TRUTH TABLE FOR THE STATEMENT FORM. (5 POINTS EACH)
A. (p ∧ ~r) ↔ (q ∨ r)
p q r ~r p ∧ ~r q ∨ r (p ∧ ~r) ↔ (q ∨ r)
T T T F
F
T
F
T T F T
T
T
T
T F T F
F
T
F
T F F T
T
F
F
F T T F
F
T
F
F T F T
F
T
F
F F T F
F
T
F
F F F T
F
F
T
B. (p → r) ↔ (q → r)
p q r p → r q → r (p → r) ↔ (q → r)
T T T
T
T
T
T T F
F
F
T
T F T
T
T
T
T F F
F
T
F
F T T
T
T
T
F T F
T
F
F
F F T
T
T
T
F F F
T
T
T
C. (p → (q → r)) ↔ ((p ∧ ~r) → r)
p q r ~r q → r p → (q → r) p ∧ ~r (p ∧ ~r) → r (p → (q → r)) ↔ ((p ∧ ~r) → r)
T T T F
T
T
F
T
T
T T F T
F
F
T
F
T
T F T F
T
T
F
T
T
T F F T
T
T
T
F
F
F T T F
T
T
F
T
T
F T F T
F
T
F
T
T
F F T F
T
T
F
T
T
F F F T
T
T
F
T
T
D. ~(~(a ∨ b ∨ c) ↔ d) → (~a ∧ ~b ∧ ~c)
a b c d ~a ~b ~c
T
T
T
T
T
T
T
T
F
F
F
F
F
F
F
F
T
T
T
T
F
F
F
F
T
T
T
T
F
F
F
F
T
T
F
F
T
T
F
F
T
T
F
F
T
T
F
F
T
F
T
F
T
F
T
F
T
F
T
F
T
F
T
F
F
F
F
F
F
F
F
F
T
T
T
T
T
T
T
T
F
F
F
F
T
T
T
T
F
F
F
F
T
T
T
T
F
F
T
T
F
F
T
T
F
F
T
T
F
F
T
T
a
∨b
∨c
~(a
∨b
∨ c)
~(a ∨ b
∨ c)
↔d
~a
∧ ~b
∧ ~c
T
T
T
T
T
T
T
T
T
T
T
T
T
T
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
T
T
F
T
F
T
F
T
F
T
F
T
F
T
F
T
T
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
T
T
(~(a ∨ b ∨ c)
↔ d)
→ (~a ∧ ~b
∧ ~c)
T
F
T
F
T
F
T
F
T
F
T
F
T
F
T
T
~(~(a ∨ b ∨ c)
↔ d)
→ (~a ∧ ~b
∧ ~c)
F
T
F
T
F
T
F
T
F
T
F
T
F
T
F
F
PART 2: USE THE TRUTH TABLE TO VERIFY THE FOLLOWING LOGICAL EQUIVALENCE (5 POINTS EACH)
A. p → (q ∨ r) ≡ (p ∧ ~q) → r
(p ∧ ~q) → r p ∧ ~q
p q r ~q (q ∨ r) p → (q ∨ r)
T T T F
T
T
≡
T
F
T T F F
T
T
≡
T
F
T F T T
T
T
≡
T
T
T F F T
F
F
≡
F
T
F T T F
T
T
≡
T
F
F T F F
T
T
≡
T
F
F F T T
T
T
≡
T
F
F F F T
F
T
≡
T
F
B. (p ∧ ~q) → r ≡ (p ∧ ~r) → q
p q r ~r p ∧ ~q (p ∧ ~q) → r
T T T F
F
T
T T F T
F
T
T F T F
T
T
T F F T
T
F
F T T F
F
T
F T F T
F
T
F F T F
F
T
F F F T
F
T
C. p → (q → r) ≡ (p → q) → r
p q r q → r p → (q → r)
T T T
T
T
T T F
F
F
T F T
T
T
T F F
T
T
F T T
T
T
F T F
F
T
F F T
T
T
F F F
T
T
≡
≡
≡
≡
≡
≢
≡
≢
≡
≡
≡
≡
≡
≡
≡
≡
(p ∧ ~r) → q p ∧ ~r
T
F
T
T
T
F
F
T
T
F
T
F
T
F
T
F
(p → q) → r p → q
T
T
F
T
T
F
T
F
T
T
F
T
T
T
F
T
p
T
T
T
T
F
F
F
F
PART 3: WRITE THE NEGATION FORM, CONTRAPOSITIVE, CONVERSE, INVERSE, NECESSARY, SUFFICIENT,
UNLESS FOR EACH FOR THE FOLLOWING STATEMENTS (21 POINTS)
A. There is an undeclared variable or there is a syntax error in the first five lines.
If-then form:
If there is a declared variable, then there is a syntax error in the first five lines.
Negation form: There is an undeclared variable and there is no syntax error in the first five lines.
Contrapositive: If there is no syntax error in the first five lines, then there is an undeclared variable.
Converse:
If there is a syntax error in the first five lines, then there is a declared variable.
Inverse:
If there is an undeclared variable, then there is no syntax error in the first five lines.
Necessary:
If there is an undeclared variable, then there is no syntax error in the first five lines.
Sufficient:
If there is a declared variable, then there is a syntax error in the first five lines.
B. If there is a syntax error in the first five lines, then there is a missing semicolon or a variable name is
misspelled.
If-then form:
If there is a syntax error in the first five lines, then there is a missing semicolon or a
variable name is misspelled.
Negation form: There is no syntax error in the first five lines and there is no missing semicolon and no
variable name is misspelled.
Contrapositive: If there is no missing semicolon or no variable name is misspelled, then there is no
syntax error in the first five lines.
Converse:
If there is a missing semicolon or a variable name is misspelled, then there is a syntax
error in the first five lines.
Inverse:
If there is no syntax error in the first five lines, then there is no missing semicolon or
no variable name is misspelled.
Necessary:
If there is no syntax error in the first five lines, then there is no missing semicolon or
no variable name is misspelled.
Sufficient:
If there is a syntax error in the first five lines, then there is a missing semicolon or a
variable name is misspelled.
C. This real number is rational or it is irrational
If-then form:
If this real number is not rational then it is irrational.
Negation form: This real number is rational and it is not irrational.
Contrapositive: If it is not irrational, then this real number is rational.
Converse:
If it is irrational, then this real number is not rational.
Inverse:
If this real number is rational, then it is not irrational.
Necessary:
If this real number is rational, then it is not irrational.
Sufficient:
If this real number is not rational then it is irrational.
PART 4: USE THE TRUTH TABLES TO DETERMINE WHETHER THE ARGUMENTS FORM IS VALID OR INVALID.
INDICATE WHICH THE COLUMNS REPRESENT THE PREMISES AND WHICH REPRESENT THE CONCLUSION AND
INCLUDE A SENTENCE THAT EXPLAINING HOW THE TRUTH TABLE SUPPORT YOUR ANSWER. (5 POINTS EACH)
A.
p∨q
p→~q
p→r
∴r
p
T
T
T
T
F
F
F
F
B.
q
T
T
F
F
T
T
F
F
r ~q p ∨ q p → ~q p → r
T F
T
F
T
F F
T
F
F
T T
T
T
T
F T
T
T
F
T F
T
T
T
F F
T
T
T
T T
F
T
T
F T
F
T
T
r
T
F
T
F
T
F
T
F
p∧q
p∨~q
~q→p
∴~r
p
T
T
T
T
F
F
F
F
q
T
T
F
F
T
T
F
F
r ~r ~q p ∧ q p ∨ ~q ~q → p
T F
F
T
T
T
F T
F
T
T
T
T F
T
F
T
T
F T
T
F
T
T
T F
F
F
F
T
F T
F
F
F
T
T F
T
F
T
F
F T
T
F
T
F
r
T
F
T
F
T
F
T
F
C.
p→(q∨r)
~q∨~r
∴~p∨~r
p
T
T
T
T
F
F
F
F
D.
q
T
T
F
F
T
T
F
F
r ~p ~q ~r p → (q ∨ r) ~q ∨ ~r ~p ∨ ~r
T F
F
F
T
F
F
F F
F
T
T
T
T
T F
T
F
T
T
F
F F
T
T
F
T
T
T T
F
F
T
F
T
F T
F
T
T
T
T
T T
T
F
T
T
T
F T
T
T
T
T
T
r∨s
~s→~t
~q∨s
∴~s
q
T
T
T
T
F
F
F
F
r
T
T
F
F
T
T
F
F
s
T
F
T
F
T
F
T
F
t ~q ~s ~t r ∨ s ~s → ~t ~q ∨ s ~s
T F
F
F
T
T
T
F
T F
T
F
T
F
T
T
T F
F
F
T
T
T
F
T F
T
F
F
F
T
T
T T
F
F
T
T
T
F
T T
T
F
T
F
F
T
T T
F
F
T
T
T
F
T T
T
F
F
F
F
T