Practice Problems Week 4

  • You may collaborate and are encouraged to collaborate with your peers. If you do, be sure to understand the material and write it out in your own words.

There is no requirement to submit the material, but this will help you solidify the concepts.

Problems

  1. Prove the following statements:

    1. Given that $ \forall x, F(x) \lor G(x)$ and $\lnot \exists x G(x)$, conclude that $\forall x, F(x)$
    2. $ \forall x (\lnot F(x) \lor G(x)) \to \forall x (F(x) \to G(x))$
    3. $\exists x F(x) \iff \lnot \forall x \lnot F(x) $

  2. Consider the statement: For all integers n, if n is even then 8n is even.

    1. Prove the statement. What kind of proof are you using?
    2. What is the converse of the statement?
    3. Prove or disprove the converse.

  3. Prove the statement: For all integers n, if 5n is odd then n is odd. Clearly state what kind of proof you are using.

  4. Prove the statement: For all real numbers x,y, $x = y$ if and only if $xy = \frac {(x+y)^2}{4}$.

    Note that you will need to prove both 'directions' of the implication.

  5. Suppose you would like to prove the following implication:

    "For all numbers n, if n is prime, then n is solitary"

    How would you start and end your argument if you tried to prove the statement...

    1. Directly
    2. By contradiction
    3. By contrapositive

    You do not have to actually prove the statement, as we haven't covered primes and solitary numbers. Focus on the structure of the argument.

  6. A friend shows you the following proof that shows the statement 1 = 3:

    • (i)Assume $ 1 = 3 $
    • (ii)Therefore $ 1 - 2 = 3 - 2 $
    • (iii)Simplifying we get $ -1 = 1 $
    • (iv)$ (-1)^2 = 1^2 $
    • (v)$ 1 = 1 $ which is true
    • (vi)Therefore $ 1 = 3 $

    Where is the flaw in the argument?

References

These problems were drawn from: