Propositional Logic

Which one are propositions?

• 3*2 = 5

• There was snow in Khon Kaen yesterday.

• He is the most handsome man.

• 2 is an odd number.

• You should get vaccine for COVID-19

Let d be “Discrete Math is difficult,” f be "Discrete Math is fun,” and n be “Disceret Math is necessary.”. Using the symbols what is the statement for "Disceret Math is fun, but it is not both difficult and necessary.

• $f\wedge\sim\left(d\wedge n\right)$

• $f\wedge d\wedge n$

• $f\wedge\sim d\wedge n$

• $f\wedge\sim d\wedge\sim n$

Let p be “x > 2,” q be “x = 2,” and r be “x > 5”. Using the defined symbols, what is the statement 2 <= x <= 5.

•  $\left(p\vee q\right)\wedge\sim r$ 

•  $\left(p\vee q\right)\wedge r$ 

•  $\left(p\wedge q\right)\wedge\sim r$ 

•  $\sim\left(p\vee q\right)\wedge r$ 

Which one are logically equivalent?

• p V (p V q) and p

• (pundefinedq)undefinedr and pundefined(q undefined r)

• pundefined(q V ~r) and (pundefinedq) V (pundefined~r)

• (pundefinedq) V r and pundefined(q V r)

Which one are logically equivalent?

• p $\rightarrow$ q and ~p V q

• ~(p $\rightarrow$ q) and p$\wedge$ ~q

• p $\rightarrow$ (q V r) and p $$\wedge (~q $\rightarrow$ r)

• p $\rightarrow$ (q $\rightarrow$ r) and (p $\rightarrow$ q) $\rightarrow$ r

What is a contrapositive of the following statement "If the world is ending tomorrow then I will not go to school"

• If I go to school then the world is not ending tomorrow.

• If I do not go to school then the world is ending tomorrow.

• If the world is not ending tomorrow then I will go to school

• If I do not go to school then the world is not ending tomorrow.

What is the negation of the following statement? "If a butterfly lays eggs then there will be more butterfiles".

• A butterfly lays eggs and there is no more butterflies.

• If a butterfly lays egg then these is no more butterflies

• If a butterfly does not lay egg then there is no more butterflies.

• If a butterfly does not lay egg then there is more butterflies

Which of the following are tautology?

• $p\ \rightarrow\left(q\ \vee\ r\right)\ \equiv\left(p\ \wedge\ \sim q\right)\ \longrightarrow\ r$

• $p\ \wedge\ \sim p$

• $\left(p\wedge q\right)\vee\left(\sim p\vee\left(p\wedge\sim q\right)\right)$

• $\left(p\wedge\sim q\right)\wedge\left(\sim p\vee q\right)$

• $\left(p\rightarrow r\right)\leftrightarrow\left(q\rightarrow r\right)$

Which of the following are contradiction ?

• $p\ \rightarrow\left(q\ \vee\ r\right)\ \equiv\left(p\ \wedge\ \sim q\right)\ \longrightarrow\ r$

• $p\ \wedge\ \sim p$

• $\left(p\wedge q\right)\vee\left(\sim p\vee\left(p\wedge\sim q\right)\right)$

• $\left(p\wedge\sim q\right)\wedge\left(\sim p\vee q\right)$

• $\left(p\rightarrow r\right)\leftrightarrow\left(q\rightarrow r\right)$

Which of the following statements are true?

• An argument is a sequence of propositions

• Each proposition before the final one is called conclusion

• The final proposition is called the premise

• The argument is valid if assuming all the premises are true then the conclusion is true.

• To validate an argument, we assume no premises are true.