Math 310 : Some Review Problems for Midterm 1
1. Prove that if A,Band Care sets such that C⊆Aand C⊆B, then C⊆A∩B
2. For all real numbers xand y, prove that |x+y|<|x|+|y|.
You may use the definition of |a|, the addition, multiplication and transitivity
laws for inequalities, and all basic arithmetic properties of real numbers.
3. Consider the following ”proof”
THEOREM: For all sets A,Band C, if C⊆A∪Band B∩C=∅, then C⊆A.
”PROOF”: Let A={a, b, c, d, e}and B={d, e, f, g}. If C⊆A∪B, then the elements
of C must be drawn from the list a, b, c, d, e, f, g. But B∩C=∅so that Band
Chave no elements in common. Therefore, the elements of Cmust, in fact, be
drawn from the list a, b, c. Since each of these elements is also an element of A, it
follows that C⊆A.
a) What is wrong with this argument?
b) Write a correct proof of this result.
4. Prove that, for all non-negative integers n, 42n+1 + 3n+2 is divisible by 13.
5. Show that the sum of the squares of the first n+ 1 non-negative integers is
n(n+ 1)(2n+ 1)
6.
6. Consider the sybolic statement ∀x∈R,∀y∈R,[(x<y)⇒x2<y2].
a) Is the statement true or false? If true, prove it. If false, give a counterex-
ample.
b) Write the symbolic negation of the statement.
7. Draw up a truth table for the statement (p⇒r)∧(r⇒q).
8. Let propositions S, W, R and T be defined as follows:
S: The sun shines.
W: The wind blows.
R: The rain falls.
T: The temperature rises.
(i) Translate into English: ¬(W∧R)⇔S
(ii) Translate into symbols: ”The sun shines and the wind doesn’t blow, and
the temperature rises only if the rain falls.”
(iii) Suppose all of S, W, R, T are true. (Yes, it’s a weird day:)) Decide which
are true: a) (S⇒W)∧(¬R∧T)b) (S∨ ¬R)⇔(T∨ ¬W)c) ¬(R∨ ¬T)∧S
9. Prove that for any sets Aand B,(A−B)∩B=∅.
10. Does the set S={1−1/n |n∈Z+}have a greatest element? Prove your answer.
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