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MATH 55 MIDTERM 1 PRACTICE, FALL 2023

(1) Determine whether each statement below is true or false, and circle (T) or (F) accordingly. You do NOT need to justify your answers.

(1a) ¬(p ↔ q) is logically equivalent to p ↔ ¬q

T F

(1b) Consider the function f(x) = x 2 whose domain and codomain are the positive real numbers. This function is a bijection.

T F

(1d) Consider the function f(x) = x 3 whose domain and codomain are the integers. This function is a bijection.

T F

(1e) The inverse of a statement is logically equivalent to the statement.

T F

(1f) The negative real numbers are countable.

(1g) ∃x∃y x2 + y 2 = 5 (where the domain is the integers).

T F

(1h) ∃x∀y 2x = y (where the domain is the real numbers).

T F

(1i) If a and b are integers and a divides b then a is odd or b is even.

T F

(1j) You can conclude that A = B if A, B, C are sets such that A∪C = B∪C.

(2) You need to justify your work

(2a) Let a = (a1, a2, a3, . . .) be the sequence defined by the recurrence

an = 2an−1 + an−2

for n > 2 and a1 = 1 and a2 = 2. What is a5?

(2b) Circle the integers that are congruent to −4 mod 7

18 35 10 24

(2c) Determine if this argument is valid. If n is a real number and n > 2 then n2 > 4. Suppose n ≤ 2 then n 2 ≤ 4.

(2d) Determine the truth value of

∀A ∀B A − B = ¯A ∩ B

(3) Prove that if x and y are real numbers then

min(x, y) = (x + y − |x − y|)/2.

where |x| is the absolute value of x (i.e |x| = x if x ≥ 0 and −x otherwise.)

(4) Let g be a function from A to B. Let S and T be subsets of A. Prove that g(S ∪ T) ⊆ g(S) ∪ g(T).