Question

1. translate the following propositions. let a, b, c, and d be the propositions: hint:¬ means

ot\a: you are doing this correctly c: you pass the examb: you get a good grade d: you see a cute puppya. a→cb. (a ∧ c)→bc. write the contrapositive of question a.d. write the converse of question b

Explanation:

Step1: Translate \(a

ightarrow c\)
The implication \(p
ightarrow q\) is translated as "If \(p\), then \(q\)". Here \(p = a\) (You are doing this correctly) and \(q = c\) (You pass the exam).
So, \(a
ightarrow c\) is "If you are doing this correctly, then you pass the exam".

Step2: Translate \((a\wedge c)

ightarrow b\)
The conjunction \(p\wedge q\) is " \(p\) and \(q\)" and the implication \(p
ightarrow q\) is "If \(p\), then \(q\)". Here \(p=a\wedge c\) (You are doing this correctly and you pass the exam) and \(q = b\) (You get a good grade).
So, \((a\wedge c)
ightarrow b\) is "If you are doing this correctly and you pass the exam, then you get a good grade".

Step3: Find the contra - positive of \(a

ightarrow c\)
The contra - positive of \(p
ightarrow q\) is \(
eg q
ightarrow
eg p\). Here \(p = a\) and \(q = c\).
So, the contra - positive is \(
eg c
ightarrow
eg a\), which is "If you do not pass the exam, then you are not doing this correctly".

Step4: Find the converse of \((a\wedge c)

ightarrow b\)
The converse of \(p
ightarrow q\) is \(q
ightarrow p\). Here \(p=(a\wedge c)\) and \(q = b\).
So, the converse is \(b
ightarrow(a\wedge c)\), which is "If you get a good grade, then you are doing this correctly and you pass the exam".

Answer:

a. If you are doing this correctly, then you pass the exam.
b. If you are doing this correctly and you pass the exam, then you get a good grade.
c. If you do not pass the exam, then you are not doing this correctly.
d. If you get a good grade, then you are doing this correctly and you pass the exam.