Problem Set 1
5 points

The problems are taken from the text. Keep in mind starred problems have solutions in the back!

Chapter 1.1
Exercises #2 (all), #3 (a, d, e, f), #6 (e, g, d, f), #8 (all), #9 (b,d), 11 (b), (d), (h)

Chapter 1.2
Exercises #1 (c,d,g), 2 (c,d,g), 3(c,d), 5 (a,b, g), 6(f,g, h,i,j), 9(b,c), 10(b),11(b), 12(a),  14(a,d), 16(b)

Some Select Solutions:
Chapter 1.1
2(i) This is a false proposition. Look at where the "that"s occur in the sentence and you will see you can rewrite it with parentheses:
It is not the case that (39 is prime, or 64 is a power of 2).
Clearly 64 is a power of 2, so the parenthetical statement is T, which means that the negation of it is F. (The final answer is "F")
2(j) is not a proposition. This is because you cannot give a truth value to the statement "this statement is false".


Chapter 1.2
2(c) The key is that this statement is "only if" not "if". The staement "A only if B" means B will not occur if A does not occur (since it will only occur if A does occur -- and even then it's not a guarantee!). For example, imagine I say "I will  go swimming only if the temperature is above 80 degree." I am not saying I will go swimming if the temp is above 80, I am saying I won't even consider if it's not. So we can rewrite this to be "if the temperature is below 80, I will not go swimming". To translate in terms of "A only if B", we rewrite "If ~B, then ~A".

So we can rewrite (c) to be
"If b does not divide 9, then b does not divide 3." Now we can find the
converse:
"If b does not divide 3, then b does not divide 9." (alternatively, "b divides 9 only if b divides 3").
and the contrapositive:
"If b divides 3, then it divides 9."

I also accepted the following re-interpretation: "A only if B" is also equivalent to "A implies B" (since the only way A can be true is if B is true). For example, looking again at the swimming example, I was also saying "If I am swimming, then it must be over 80 degress outside."
From this view, the converse is then "B implies A." and the converse "~B implies ~A".

So we rewrite (c) to be
"If b divides 3, then b divides 9."
and the converse:
"If b divides 9, then b divides 3."
and the contrpositive:
"If b does not divide 9, then it does not divide 3."

2(g) The statement can be rewritten by understanding "necessary" to mean that it has to be there in order for the other thing to be true. So 1+2=3 is necessary for 1+1=2 means that "If 1+1=2, it must have been the case that 1+2=3". In other words:
If 1+1=2, then 1+2=3.
The converse is therefore
If 1+2=3, then 1+1=2.
The contrapositive is
If 1+2 is not 3, then 1+1 is not 2.

9(b) Not a definition. There are a lot of functions that are not quadratic and have an x^2 term, like f(x) =x^2 + ln(x). A good definition of y=f(x) is a quadratic function is if it can be written in the form f(x) = ax^2 + bx + c for some constants a,b,c.
(c) This is a definition.

12(a) This can be done, as many of you did, by writing out a truth table and comparing. Be sure to finish your proof by saying, "The two propositions are equivalent because they have the same truth values."

A better/easier/more concise proof is the following:
The contrapositive of "(P or Q) implies R" is the statement, "~R implies ~(P or Q)". By DeMorgan's laws, ~(P or Q) is the same as ~P and ~Q. Therefore, the first proposition is equivalent to "~R implies (~P and ~Q)", which is the second proposition.