Discrete Math: Logic and Proofs

Given the conditional statement "If I am hungry, then I will eat pizza," write the converse, the inverse, and the contrapositive of the statement.

Prove that there is no way to have a lossless compression algorithm that always produces a smaller output for any given input.

Prove that a given relation $R$ on the set of integers $\mathbb\Z$ is reflexive, symmetric, and transitive.

For all x in the domain D, the predicate P(x) is true.

There exists an X in the domain such that the predicate is true.

For all $x$, there exists a $y$ such that $x - y = 0$. Is this statement true or false?

For all $x$, there exists a $y$ such that $3x - y = 0$. Is this statement true or false?

For all $x$, there exists a $y$ such that $x - 3y = 0$. Is this statement true or false?

There exists an $x$ and there exists a $y$ such that $x + y = 15$. Is this statement true or false?

For all $x$, $x > 5$ implies that for all $y$ such that $y > x$, $y > 5$. Is this statement true or false?

For all $x$, $x > 5$ implies that for all $y$ such that $y > x$, $y > 7$. Is this statement true or false?

Determine if a proposition is a tautology, contradiction, or contingency using logical equivalences.

Verify the logical equivalence $\sim (\sim p \land q) \land (p \lor q) = p$ using the laws of logic.

Negate the conditional statement $ ext{P} ightarrow ext{Q}$.

Negate the statement: "If it is raining, then it is cloudy."

Negate the statement: "If it is a toaster, then it is made of gold."

Negate the statement: "If wishes were horses, then beggars would ride."

Negate the statement: "If it is blue then it is not spinach."