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RULE OF INFERENCE
B. Determine if the following argument is valid. Explain by using rule of
inference.
1. If you perform every programming problem in the module, then you
will learn programming. You learned programming. Therefore, you
perform every programming problem in the module.
2. Not everyone likes to go to the hospital; hence, there is someone
who does not like to go to the hospital.
RULE OF INFERENCE
A. What rule of inference is used in each of the following arguments?
Show solution.
1. If I will read my modules, then I can answer all the activities. If I can
answer all the activities, then I will get high scores. Therefore, if I will
read my modules, then I will get high scores.
2. Rizza is an IT student. Therefore, Rizza is either an IT student or a
programmer.
3. If it is national holiday, then school is closed. It is national holiday.
Therefore, the school is closed.
4. If Ann does not love numbers or if Ann does not love programming.
If Ann loves numbers, then she can be a mathematician. Therefore,
Ann can be a mathematician.
B.Let p, q and r denotes the following statements:
p: A square has four equal side
q: Rectangle has 2 parallel sides
r: A square is a rectangle.
1. Express each of the following into English sentence. (3 pts each)
a. r ^ q → p
b. p
̅ → q
c. q → p
̅ v r
2. Write T if the above item is true and F if it false. Show solution. (3 pts
each)
a.
b.
c.
C. Show whether or not p ↔ q ≡ (p → q) ^ (q → p) (8 pts)
D. Find the converse, inverse and contrapositive of the implication: “If today is
Monday then, I have an exam today.” (3 pts each)
1. Inverse:
2. Converse:
3. Contrapositive:
Translate the following English sentence into symbol. (3 pts each)
1. No one in this class is wearing pants and a guitarist.
Let:
Domain of x is all persons
A(x): x is wearing pants
B(x): x is a guitarist
C(x): belongs to the class
Answer: __________________________________________________
2. No one in this class is wearing pants and a guitarist.
Let:
Domain of x is persons in this class
A(x): x is wearing pants
B(x): x is a guitarist
Answer: __________________________________________________
3. There is a student at your school who knows C++ but who doesn’t
know Java.
Let:
Domain: all students at your school
C(x): x knows C++
J(x): x knows Java
Answer: __________________________________________________
B. Write the following predicates symbolically and determine its truth value.
Note: Use at least three (3) values for the variables. (5 pts each)
1. for every real number x, if x>1 then x – 1 > 1
2. for some real number x, x2 ≤ 0
C. Translate the following English sentence into symbol. (3 pts each)
1. No one in this class is wearing pants and a guitarist.
Let:
Domain of x is all persons
A(x): x is wearing pants
B(x): x is a guitarist
C(x): belongs to the class
2. No one in this class is wearing pants and a guitarist.
Let:
Domain of x is persons in this class
A(x): x is wearing pants
B(x): x is a guitarist
3. There is a student at your school who knows C++ but who doesn’t
know Java.
Let:
Domain: all students at your school
C(x): x knows C++
J(x): x knows Java
PREDICATE LOGIC.(25 pts)
A. Let P(x) be the statement x
2 > x4. If the domain consists of the integers,
what are the truth values?
1. P(0)
2. P(-1)
3. P(1)
4. P(2)
5. ∃xP(x)
6. ∀xP(x)
Using a Truth table, determine the value of the compound proposition ((𝑝 ∨ 𝑞) ∧ (￢𝑝 ∨ 𝑟)) → (𝑞 ∨ 𝑟).
PREDICATE LOGIC.(25 pts)
A. Let P(x) be the statement x 2 > x4. If the domain consists of the integers,
what are the truth values?
1. P(0)
2. P(-1)
3. P(1)
4. P(2)
5. ∃xP(x)
6. ∀xP(x)
B. Write the following predicates symbolically and determine its truth value.
Note: Use at least three (3) values for the variables. (5 pts each)
1. for every real number x, if x>1 then x – 1 > 1
2. for some real number x, x2 ≤ 0
C. Translate the following English sentence into symbol. (3 pts each)
1. No one in this class is wearing pants and a guitarist.
Let:
Domain of x is all persons
A(x): x is wearing pants
B(x): x is a guitarist
C(x): belongs to the class
2. No one in this class is wearing pants and a guitarist.
Let:
Domain of x is persons in this class
A(x): x is wearing pants
B(x): x is a guitarist
3. There is a student at your school who knows C++ but who doesn’t
know Java.
Let:
Domain: all students at your school
C(x): x knows C++
J(x): x knows Java
PROPOSITIONAL LOGIC:
Let p, q and r denotes the following statements:
p: A square has four equal side
q: Rectangle has 2 parallel sides
r: A square is a rectangle.
1. Express each of the following into English sentence. (3 pts each)
a. r ^ q → p
b. p
̅ → q
c. q → p
̅ v r
2. Write T if the above item is true and F if it false. Show solution. (3 pts
each)
a.
b.
c.
C. Show whether propositional or not p ↔ q ≡ (p → q) ^ (q → p) (8 pts)
D. Find the converse, inverse and contrapositive of the implication: “If today is
Monday then, I have an exam today.” (3 pts each)
1. Inverse:
2. Converse:
3. Contrapositive:
An urn contains numbered balls. How many ways can we choose balls out of the urn (without repetition, the order does not count)6
63
