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COMP9020 2024 Term 2 - Problem Set 2 of 8 Deadline - Monday June 17, 09:00. Ans...

Jun 16, 2024

Solution by Steps

step 1

Let

H(x)

represent "x is hungry" and let

A

represent Alice

step 2

The statement "Either Alice is hungry or everyone is hungry" can be written as:

H(A) \lor \forall x \, H(x)

Answer

H(A) \lor \forall x \, H(x)

Key Concept

Predicate Logic Translation

Explanation

We use

H(x)

to denote "x is hungry" and

A

for Alice, then translate the statement directly.

Question 2 Translate the following into fully symbolic predicate logic - "If someone is tall then Alice is tall".

step 1

Let

T(x)

represent "x is tall" and let

A

represent Alice

step 2

The statement "If someone is tall then Alice is tall" can be written as:

\exists x \, T(x) \rightarrow T(A)

Answer

\exists x \, T(x) \rightarrow T(A)

Key Concept

Predicate Logic Translation

Explanation

We use

T(x)

to denote "x is tall" and

A

for Alice, then translate the statement directly.

Question 3 Translate the following into fully symbolic predicate logic - "If all students are studying hard, then everyone is happy".

step 1

Let

S(x)

represent "x is studying hard" and

H(x)

represent "x is happy"

step 2

The statement "If all students are studying hard, then everyone is happy" can be written as:

\forall x \, S(x) \rightarrow \forall x \, H(x)

Answer

\forall x \, S(x) \rightarrow \forall x \, H(x)

Key Concept

Predicate Logic Translation

Explanation

We use

S(x)

to denote "x is studying hard" and

H(x)

to denote "x is happy", then translate the statement directly.

Question 4 Is the following wff a logical truth? If so, then justify your answer in your own words. If not, then provide a countermodel: -

\exists x P x \leftrightarrow (\exists x P x \rightarrow \exists x Q x)

step 1

Analyze the given wff:

\exists x P x \leftrightarrow (\exists x P x \rightarrow \exists x Q x)

step 2

The left side

\exists x P x

asserts the existence of some

x

such that

P(x)

is true

step 3

The right side

(\exists x P x \rightarrow \exists x Q x)

asserts that if there exists some

x

such that

P(x)

is true, then there exists some

x

such that

Q(x)

is true

step 4

\exists x P x

is true, then

(\exists x P x \rightarrow \exists x Q x)

must also be true if

\exists x Q x

is true

step 5

\exists x P x

is false, then

(\exists x P x \rightarrow \exists x Q x)

is true by the definition of implication

step 6

Therefore,

\exists x P x \leftrightarrow (\exists x P x \rightarrow \exists x Q x)

is always true, making it a logical truth

Answer

Logical Truth

Key Concept

Logical Equivalence

Explanation

The given wff is always true regardless of the truth values of

P(x)

and

Q(x)

, making it a logical truth.

Question 5 Is the following argument valid? If so, justify your answer in your own words. If not, then provide a countermodel:

\text { 1. } \begin{aligned} \forall x &amp; (P x \rightarrow Q x) \\ \text { 2. } &amp; \sim Q a \\ \therefore &amp; \sim P a \end{aligned}

step 1

The first premise

\forall x (P(x) \rightarrow Q(x))

states that for all

x

, if

P(x)

is true, then

Q(x)

must also be true

step 2

The second premise

\sim Q(a)

states that

Q(a)

is false

step 3

To determine the validity, we need to check if

\sim P(a)

logically follows from the premises

step 4

From

\forall x (P(x) \rightarrow Q(x))

, we know that if

P(a)

were true, then

Q(a)

would have to be true

step 5

Since

Q(a)

is false,

P(a)

cannot be true (otherwise it would contradict the first premise)

step 6

Therefore,

\sim P(a)

must be true, making the argument valid

Answer

Valid

Key Concept

Modus Tollens

Explanation

The argument uses the logical rule of Modus Tollens, which states that if

P \rightarrow Q

and

\sim Q

are true, then

\sim P

must also be true.