Logika Matematika Yang Kamu Anggap Benar

1. Diketahui pernyataan : I Jika hari panas, maka Ani memakai topi II Ani tidak memakai topi atau ia memakai payung III Ani tidak memakai payung Kesimpulan yang sah adalah ….

Hari panas

Hari tidak panas

Ani memakai topi

Hari panas dan Ani memakai topi

Hari tidak panas dan Ani memakai topi

The valid conclusion is "Ani memakai topi" because the statement I states that if it is a hot day, then Ani wears a hat. Since it is stated that it is not a hot day, the conclusion can be drawn that Ani wears a hat. The other options do not follow from the given statements.

Explanation

The valid conclusion is "Ani memakai topi" because the statement I states that if it is a hot day, then Ani wears a hat. Since it is stated that it is not a hot day, the conclusion can be drawn that Ani wears a hat. The other options do not follow from the given statements.

2. Diketahui premis berikut : I. Jika Budi rajin belajar maka ia menjadi pandai. II. Jika Budi menjadi pandai maka ia lulus ujian. III. Budi tidak lulus ujian. Kesimpulan yang sah adalah ….

Budi menjadi pandai

Budi rajin belajar

Budi lulus ujian

Budi tidak pandai

Budi tidak rajin belajar

Based on the given premises, the conclusion that can be drawn is that Budi is not diligent in studying. This can be inferred from the second premise which states that if Budi becomes smart, then he passes the exam. Since the third premise states that Budi did not pass the exam, it can be concluded that Budi did not become smart, which implies that he was not diligent in studying.

Explanation

Based on the given premises, the conclusion that can be drawn is that Budi is not diligent in studying. This can be inferred from the second premise which states that if Budi becomes smart, then he passes the exam. Since the third premise states that Budi did not pass the exam, it can be concluded that Budi did not become smart, which implies that he was not diligent in studying.

3. Ditentukan premis – premis :

Jika Badu rajin bekerja maka ia disayang ibu.
Jika Badu disayang ibu maka ia disayang nenek
Badu tidak disayang nenek

Kesimulan yang sah dari ketiga premis diatas adalah …

Badu rajin bekerja tetapi tidak disayang ibu

Badu rajin bekerja

Badu disayang ibu

Badu disayang nenek

Badu tidak rajin bekerja

Based on the given premises, the conclusion "Badu tidak rajin bekerja" is valid. The first premise states that if Badu is diligent, then his mother loves him. The second premise states that if Badu is loved by his mother, then his grandmother loves him. The third premise states that Badu is not loved by his grandmother. Therefore, we can conclude that Badu is not diligent in his work because if he were diligent, his mother would love him and consequently his grandmother would also love him.

Explanation

Based on the given premises, the conclusion "Badu tidak rajin bekerja" is valid. The first premise states that if Badu is diligent, then his mother loves him. The second premise states that if Badu is loved by his mother, then his grandmother loves him. The third premise states that Badu is not loved by his grandmother. Therefore, we can conclude that Badu is not diligent in his work because if he were diligent, his mother would love him and consequently his grandmother would also love him.

4. Kontraposisi dari pernyataan majemuk p → ( p V ~q ) adalah ….

( p V ~q ) → ~p

(~p Λ q ) → ~p

( p V ~q ) → p

(~p V q ) → ~p

( p Λ ~q ) → ~p

The correct answer is (~p Λ q ) → ~p. This is the contrapositive of the given compound statement. In a contrapositive, the antecedent and the consequent are negated and swapped. In this case, the original statement is "p → (p V ~q)", so the contrapositive is "~(p V ~q) → ~p". By De Morgan's law, ~(p V ~q) is equivalent to (~p Λ q), so the correct answer is (~p Λ q) → ~p.

Explanation

The correct answer is (~p Λ q ) → ~p. This is the contrapositive of the given compound statement. In a contrapositive, the antecedent and the consequent are negated and swapped. In this case, the original statement is "p → (p V ~q)", so the contrapositive is "~(p V ~q) → ~p". By De Morgan's law, ~(p V ~q) is equivalent to (~p Λ q), so the correct answer is (~p Λ q) → ~p.

5. Invers dari pernyataan p → ( p Λ q )

(~p Λ ~q ) → ~p

(~p V ~q ) → ~p

~p → (~p Λ ~q )

~p → (~p Λ q )

~p → (~p V ~q )

The correct answer is ~p → (~p V ~q). This answer is derived by applying the rule of implication, which states that if we have a conditional statement of the form "p → q", then the negation of the antecedent (~p) implies the negation of the consequent (~q). In this case, the original statement is "p → (p ∧ q)", so by applying the rule of implication, we can conclude that ~p implies ~(p ∧ q). By using De Morgan's law, we can further simplify ~(p ∧ q) to (~p ∨ ~q). Therefore, the correct answer is ~p → (~p ∨ ~q).

Explanation

The correct answer is ~p → (~p V ~q). This answer is derived by applying the rule of implication, which states that if we have a conditional statement of the form "p → q", then the negation of the antecedent (~p) implies the negation of the consequent (~q). In this case, the original statement is "p → (p ∧ q)", so by applying the rule of implication, we can conclude that ~p implies ~(p ∧ q). By using De Morgan's law, we can further simplify ~(p ∧ q) to (~p ∨ ~q). Therefore, the correct answer is ~p → (~p ∨ ~q).

6.

Diketahui argumentasi :
1. p → q

~p ---------- ~q

p → q

~q V r ---------- p → r

p → q

p → r ---------- q → r Argumentasi yang sah adalah ….

A. saja

II saja

III saja

I dan II saja

II dan III saja

The correct answer is "II saja" because the argument only includes the premises p → q and ~q ∨ r, and the conclusion q → r. The argument does not include the premise ~p, so it cannot be used to derive the conclusion ~q. Therefore, the only valid part of the argument is II, which shows that if q is true, then r must also be true.

Explanation

The correct answer is "II saja" because the argument only includes the premises p → q and ~q ∨ r, and the conclusion q → r. The argument does not include the premise ~p, so it cannot be used to derive the conclusion ~q. Therefore, the only valid part of the argument is II, which shows that if q is true, then r must also be true.

7. Penarikan kesimpulan dengan menggunakan modus tolens didasarkan atas suatu pernyataan majemuk yang selalu berbentuk tautologi untuk setiap kasus. Pernyataan yang dimaksud adalah ….

( p → q ) Λ p → q

( p → q ) Λ ~q → ~p

( p → q ) Λ p → ( p Λ q )

( p → q ) Λ ( q → r ) → ( p → r )

( p → q ) Λ ( p → r ) → ~ ( q → r )

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Explanation

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8. Kesimpulan dari premis berikut merupakan …. p → ~q q V r ---------- p → r

Konvers

Kontra posisi

Modus ponens

Modus tollens

Silogisme

The given premises are "p implies not q" and "q or r". From these premises, we can conclude that "p implies r" using the rule of syllogism. The rule of syllogism states that if we have two conditional statements, such as "if A then B" and "if B then C", we can infer the conclusion "if A then C". In this case, the first premise "p implies not q" can be written as "if p then not q", and the second premise "q or r" can be written as "if q then r". By applying the rule of syllogism, we can conclude that "if p then r".

Explanation

The given premises are "p implies not q" and "q or r". From these premises, we can conclude that "p implies r" using the rule of syllogism. The rule of syllogism states that if we have two conditional statements, such as "if A then B" and "if B then C", we can infer the conclusion "if A then C". In this case, the first premise "p implies not q" can be written as "if p then not q", and the second premise "q or r" can be written as "if q then r". By applying the rule of syllogism, we can conclude that "if p then r".

9. Penarikan kesimpulan yang sah dari argumen tasi berikut : ~p → q q → r ---------- ....... …

P Λ r

~p V r

P Λ ~r

~p Λ r

P V r

The valid conclusion that can be drawn from the given argument is that either p or r is true, or both p and r are true. This can be represented as p V r.

Explanation

The valid conclusion that can be drawn from the given argument is that either p or r is true, or both p and r are true. This can be represented as p V r.

10. Penarikan kesimpulan yang sah dari argumentasi berikut : Jika Siti sakit maka dia pergi ke dokter Jika Siti pergi ke dokter maka dia diberi obat. adalah ….

Siti tidak sakit atau diberi obat

Siti sakit atau diberi obat

Siti tidak sakit atau tidak diberi obat

Siti sakit dan diberi obat

Siti tidak sakit dan tidak diberi obat

The valid conclusion from the given argument is that Siti is either not sick or she is given medicine. This conclusion can be drawn by combining the two statements in the argument. If Siti is sick, then she goes to the doctor. And if she goes to the doctor, then she is given medicine. Therefore, if Siti is not sick, it is still possible that she goes to the doctor and is given medicine.

Explanation

The valid conclusion from the given argument is that Siti is either not sick or she is given medicine. This conclusion can be drawn by combining the two statements in the argument. If Siti is sick, then she goes to the doctor. And if she goes to the doctor, then she is given medicine. Therefore, if Siti is not sick, it is still possible that she goes to the doctor and is given medicine.