Mathematical Logic

Let the statement $p$ be “1 is a positive number”, the statement $q$ be “$-1>0$”, and the statement $r$ be “1 is an odd number”. Is the statement “$\lnot((p\lor q)\land r)$ is false” true?

Let $n$ be an integer. Let $p$ be the statement “$n$ is a multiple of 5” and $q$ be the statement “the last digit of $n$ is 0 or 5”. Let the statement $r=(p\iff q)$.

1. Write the statement $r$ in English.

2. Is the statement $r$ true?

Simplify the statement

into a statement that uses only three operators in total.

Convert the statement “for all positive integers $n>2$ there exist integers $a$ and $b$ such that $a^{3}+b^{4}=n^{5}$” into symbolic notation using quantifiers and logical operators.

Consider the statement “there exists a real number $y$ such that $xy=1$ for all non-zero real numbers $x$”.

1. Write down two predicates $P(x)$ and $Q(x)$ in symbols such that the above statement is “$\forall x,(P(x)\implies Q(x))$”.

2. Negate the above statement, writing your answer in symbolic notation.

Which of the following are statements? Which are predicates?

1. 4 equals 5.

2. $n\geq 4$.

3. $8=2\times 4$.

4. $x^{2}-1=(x-1)(x+1)$.

5. There is a prime number $p$ that is one less than a cube and two less than a square.

6. $P(x)\equiv P(x)$.

7. $P(x)\iff P(x)$ for all $x$.

For the following, assume $p$ and $q$ are statements. Which of the following are true statements?

1. $\forall p\;\forall q\;(p\equiv q)$.

2. $\forall p\;\exists q\;(p\equiv q)$.

3. $\exists p\;\forall q\;(p\equiv q)$.

4. $\exists p\;\exists q\;(p\equiv q)$.

5. $\forall p\;\forall q\;(p\not\equiv q)$.

6. $\forall p\;\exists q\;(p\not\equiv q)$.

7. $\exists p\;\forall q\;(p\not\equiv q)$.

8. $\exists p\;\exists q\;(p\not\equiv q)$.