Question

A. Let p and q be propositions

p: 4 is a rational number

q: 3 is an irrational number.

Express each of these propositions as an English sentence:

1. ¬p =

2. p v q =

3. p ^ q =

4. p→q =

5. p ↔q =

Answer 1

Statement $p$ is true. Statement $q$ is false.

1. $\lnot p$ is a negation of $p$. Statement $\lnot p$ is: $4$ is an irrational number. This statement is false.
2. $p\lor q$ is disjunction of $p$ and $q$. The statement $p\lor q$ is: 4 is a rational number or 3 is an irrational number. The latter statement is true.
3. $p\land q$ is a conjunction of $p$ and $q$. The statement $p\land q$ is: 4 is a rational number and 3 is an irrational number. The latter statement is false.
4. $p\rightarrow q$ is a material implication from $p$ to $q$. The statement $p\rightarrow q$ is: From the fact that 4 is a rational number it follows that $3$ is an irrational number. This statement is false.
5. $p\leftrightarrow q$ is the logical biconditional. The statement $p\leftrightarrow q$ is: The statement "4 is a rational number" holds if and only if the statement "3 is an irrational number" holds. The latter statement is false.