Mathematical Statements

Suppose $P$ and $Q$ are the statements: $P$: Jack passed math. $Q$: Jill passed math.

1. Translate “Jack and Jill both passed math” into symbols.

2. Translate “If Jack passed math, then Jill did not” into symbols.

3. Translate “$P \vee Q$ ” into English.

4. Translate “$\neg(P \wedge Q) \imp Q$ ” into English.

5. Suppose you know that if Jack passed math, then so did Jill. What can you conclude if you know that:

   1. Jill passed math?
   2. Jill did not pass math?

Write each of the following statements in the form, “ if …, then ….” Careful, some of the statements might be false (which is alright for the purposes of this question).

1. To lose weight, you must exercise.

2. To lose weight, all you need to do is exercise.

3. Every American is patriotic.

4. You are patriotic only if you are American.

5. The set of rational numbers is a subset of the real numbers.

6. A number is prime i…

For a given predicate $P(x)$, you might believe that the statements $\forall x P(x)$ or $\exists x P(x)$ are either true or false. How would you decide if you were correct in each case? You have four choices: you could give an example of an element $n$ in the domain for which $P(n)$ is true or for which $P(n)$ if false, or you could argue that no matter what $n$ is, $P(n)$ is true or is false.

1. What would you need to do to prove $\forall x P(x)$ is true?

2. What would yo…

Translate into symbols. Use $E(x)$ for “$x$ is even” and $O(x)$ for “$x$ is odd.”

1. No number is both even and odd.

2. One more than any even number is an odd number.

3. There is prime number that is even.

4. Between any two numbers there is a third number.

5. There is no number between a number and one more than that number.

Translate into English:

1. $\forall x (E(x) \imp E(x +2))$.
2. $\forall x \exists y (\sin(x) = y)$.
3. $\forall y \exists x (\sin(x) = y)$.
4. $\forall x \forall y (x^3 = y^3 \imp x = y)$.