MTH 231 Study Guide - Midterm Guide: Mathematical Induction

Math 231 Midterm Practice
1. Fill in the blank (in words) to finish the following definitions:
(a) Set Xis a subset of set Yif .
(b) The union of two sets X, Y is the set consisting of elements from X Y .
(c) The intersection of two sets X, Y is the set consisting of elements from X
Y.
(d) A proposition function is a statement P(x) depending on a domain of discourse
D, such that for each x∈D.
2. Let A, B be sets in a universal set U, let p, q be propositions, and let P(x), P (x, y) be
a propositional functions on domain of discourse D. State whether the following are
TRUE or FALSE.
(a) A∪B=A∪B
(b) ¬p∧ ¬q≡ ¬(p∨q)
(c) ¬(∀xP (x)) ≡ ∀x¬P(x)
(d) ¬(∀x∃yP (x, y)) ≡ ∃x∀y¬P(x, y)
3. Let U={a, b, c, d, e, f, g, h, i, j, k}be a universal set. Let A={a, e, i},B={a, d, g, j},
and C={i, j}. Compute the following sets:
(a) B−C=
(b) B∩(A∪C) =
(c) A×C=
4. Consider the following propositions:
p: You have taste buds
q: You love dessert
r: You think ice cream is better than dinner
(a) Formulate the proposition “If you have taste buds and think that ice cream is
better than dinner, then you love dessert.” in symbols.
(b) Formulate the proposition q∨p→rin words.
(c) Use DeMorgan’s laws to reformulate the proposition ¬(p∨q) in both symbols and
words.
5. Let the domain of discourse D=R. Determine the truth value for each of the following
propositions. Give a short justification for your answers.
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