CSCI 301 'Quiz' 2

  1. Complete the missing entries in this partially filled truth table:

    \(P\) \(Q\) \(P \Rightarrow Q\) \(Q \lor P\) \((P \Rightarrow Q) \lor (Q \lor P)\)
    T T
    T F
    F T
    F F

  2. For the following pair of logical expressions, determine whether they are logically equivalent; justify your answer with a truth table or, if they are equivalent you may justify with a sequence of manipulations using known equivalences. \((\neg P \land Q) \lor (P \land \neg Q)\) and \(\neg (P \Leftrightarrow Q)\)

  3. Are the following two statements logically equivalent? Briefly explain.

    1. 4 divides \(a\) if \(a\) is even

    2. if \(a\) is not even, it is not divisible by 4

  4. Translate each statement into symbolic form using quantifiers:

    1. Some integers are perfect squares

    2. For every positive real number, there exists a smaller positive real number

    3. Every prime number greater than 2 is odd

  5. Translate each symbolic statement into English, and indicate whether they are true or false:

    1. \(\forall x \in \mathbb{Z}, \exists y \in \mathbb{Z}, x = 2y \lor x = 2y + 1\)

    2. \(\exists x \in \mathbb{R}, \forall y \in \mathbb{R}, x \cdot y = y\)

  6. Translate each statement to symbols, negate it, simplify the negation, and express the result in English:

    1. All natural numbers are positive

    2. There exists a real number that is both positive and negative

    3. \(a\) is positive and \(b\) is negative.