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Activity 1.2

1. The document provides examples of different logical implications and their truth values based on the truth values of their premises and conclusions. It examines implications where the premise and conclusion are both true or false, as well as examples where the premise is true but the conclusion is false. 2. Various logical connectives and operations are also evaluated based on the truth values of variables P, Q, and R including conjunction, disjunction, material implication, biconditional, exclusive or. 3. Converses, contrapositives and inverses of logical implications are identified. Tautologies, contingencies and absurdities are determined by examining the truth tables of complex logical statements involving multiple connectives.

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willem gogh
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16 views4 pages

Activity 1.2

1. The document provides examples of different logical implications and their truth values based on the truth values of their premises and conclusions. It examines implications where the premise and conclusion are both true or false, as well as examples where the premise is true but the conclusion is false. 2. Various logical connectives and operations are also evaluated based on the truth values of variables P, Q, and R including conjunction, disjunction, material implication, biconditional, exclusive or. 3. Converses, contrapositives and inverses of logical implications are identified. Tautologies, contingencies and absurdities are determined by examining the truth tables of complex logical statements involving multiple connectives.

Uploaded by

willem gogh
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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A.

Give examples of or tell why no such example exists:

1. A false implication with a false conclusion.

-A false implication with a false conclusion cannot exist, since a false implication with a false
conclusion would result in a true implication.

3. A true implication with a false conclusion.

P: If you are going to study. *P is true and Q is false. Therefore, the


implication is true.
Q: Then you will have a bad grade.

-If you are going to study, then you will have a bad grade.

5. A false implication with true hypothesis.

P: If you are doing great. *P is false and Q is true. Therefore, the


implication is false.
Q: Then you will be sad.

-If you are doing great, then you will be sad.

C. Give the truth values of the following. Assume that P is true, Q is false, and R is true.

1 1. 𝑝 → (𝑞 → 𝑟)

p q r (q→r) p→(q→r)
1 1 1 1 1
1 1 0 0 0
1 0 1 1 1
1 0 0 1 1
0 1 1 1 1
0 1 0 0 1
0 0 1 1 1
0 0 0 1 1

1 3.𝑝 ∧ (𝑞 → 𝑟)

p q r (q→r) p∧(q→r)
1 1 1 1 1
1 1 0 0 0
1 0 1 1 1
1 0 0 1 1
0 1 1 1 0
0 1 0 0 0
0 0 1 1 0
0 0 0 1 0
1 5.(𝑝 → 𝑞) ⟷ (~𝑞 → ~𝑝)

p q ~p ~q p→q ~q→~p (p→q)⟷(~q→~p)


1 1 0 0 1 0 1
1 0 0 1 0 0 1
0 1 1 0 1 1 1
0 0 1 1 1 1 1

1 7.[𝑝 → (𝑞 ∧ 𝑟)] ⊕ [𝑝 ∧ (𝑞 → 𝑟)]

p q r (q∧r) [p→(q∧r)] (q→r) [p∧(q→r)] [p→(q∧r)]⊕[p∧(q→r)]


1 1 1 1 1 1 1 0
1 1 0 0 0 0 0 0
1 0 1 0 0 1 1 1
1 0 0 0 0 1 1 1
0 1 1 1 1 1 0 1
0 1 0 0 1 0 0 1
0 0 1 0 1 1 0 0
0 0 0 0 1 1 0 0

1 9.(𝑝 ∨∼ 𝑞) ∧ (𝑞 → 𝑟)

p q r ~q (p∨∼q) (q→r) (p∨∼q) ∧(q→r)


1 1 1 0 1 1 1
1 1 0 0 1 0 0
1 0 1 1 1 1 1
1 0 0 1 1 1 1
0 1 1 0 0 1 0
0 1 0 0 0 0 0
0 0 1 1 1 1 1
0 0 0 1 1 1 1

D. Write the converse, contrapositive and inverse of each of the following implications.

1. If a number has a factor of 2, then it has a factor 4.

Converse: If it has a factor of 4, then a number has a factor of 2.

Contrapositive: If it doesn’t have a factor 4, then the number doesn’t have a factor of 2

Inverse: If the number doesn’t have a factor of 2, then it doesn’t have a factor 4.
E. By examining
truth tables, determine
whether each of the
following is a tautology, a contingency, or an absurdity.

TAUTOLOGY 1.(∼ 𝑝 ∧ 𝑞)⨁(𝑞 → 𝑝)

q ∼p (∼p∧q) (q→p) (∼p∧q)⨁(q→p)


1 0 0 1 1
0 0 0 1 1
1 1 1 0 1
0 1 0 1 1

TAUTOLOGY 3.(𝑝⨁𝑞) ↔ (~𝑞⨁~𝑝)

p q ~p ~q (p⨁q) (~q⨁~p) (p⨁q)↔(~q⨁~p)


1 1 0 0 0 0 1
1 0 0 1 1 1 1
0 1 1 0 1 1 1
0 0 1 1 0 0 1

CONTINGENCY 5. (𝑝 → 𝑞)⨁(𝑞 → 𝑝)

p q (p→q) (q→p) (p→q)⨁(q→p)


1 1 1 1 0
1 0 0 1 1
0 1 1 0 1
0 0 1 1 0

TAUTOLOGY 7.[(𝑝 ↔ 𝑞) ↔ 𝑟] ↔ [𝑝⨁(𝑞⨁𝑟)]

p q r (p↔q) [(p↔q)↔r] (q⨁r) [p⨁(q⨁r)] [(p↔q)↔r]↔[p⨁(q⨁r)]


1 1 1 1 1 0 1 1
1 1 0 1 0 1 0 1
1 0 1 0 0 1 0 1
1 0 0 0 1 0 1 1
0 1 1 0 0 0 0 1
0 1 0 0 1 1 1 1
0 0 1 1 1 1 1 1
0 0 0 1 0 0 0 1
CONTINGENCY 9.(∼ 𝑟⨁𝑝) ∧ (𝑝 ↔ 𝑞)

p q r ∼r (∼r⨁p) (p↔q) (∼r⨁p)∧(p↔q)


1 1 1 0 1 1 1
1 1 0 0 1 0 0
1 0 1 0 0 0 0
1 0 0 0 0 1 0
0 1 1 1 0 1 0
0 1 0 1 0 0 0
0 0 1 1 1 0 0
0 0 0 1 1 1 1

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