Exam 1

Use De Morgan's law to select the statement that is logically equivalent to: "It is not true that there was a student who was absent yesterday." A) Every student was not absent yesterday. B) At least one student was not absent yesterday. C) Some student was absent yesterday. D) Every student was absent yesterday.

A) Every student was not absent yesterday.

Graph G is defined by the arrow diagram below. What is the out-degree of vertex 2?

2

A={x∈Z:x is a prime number} B={4,7,9,11,13,14} C={x∈Z:3≤x≤10} Express the set corresponding to (A∪B)∩C(A∪B)∩C in roster notation.

(3, 4, 5, 7, 9}

The domain for variable x is the set {Ann, Ben, Cam, Dave}. The table below gives the values of predicates P and Q for every element in the domain. Name | P(x) | Q(x) Ann F F Ben T F Cam T T Dave T T Select the statement that is true. A) ∀x(P(x)→Q(x)) B )∀x(Q(x)→P(x)) C) ∀x(P(x)∧Q(x)) D) ∀x(P(x)∨Q(x))

B )∀x(Q(x)→P(x))

f : Z → Z . f ( x ) = ⌈ x / 3 ⌉ Select the correct description of the function f. A) One-to-one and onto B) Onto but not one-to-one C) One-to-one but not onto D) Neither one-to-one nor onto

B) Onto but not one-to-one

Select the relation that is an equivalence relation. The domain is the set {1, 2, 3, 4}. A) { (3, 4), (4, 3), (1, 3), (3, 1), (1, 1), (2, 2), (3, 3), (4, 4) } B) { (1, 4), (4, 1), (1, 1), (2, 2), (3, 3), (4, 4) } C) { (1, 4), (4, 1), (2, 2), (3, 3) } D) { (1, 4), (4, 1), (1, 3), (3, 1), (2, 2) }

B) { (1, 4), (4, 1), (1, 1), (2, 2), (3, 3), (4, 4) }

Which statement is false? A) 7 | 0 B) 4 | − 16 C) 2 ∤ 5 D) 1 ∤ 5

C) 2 ∤ 5

The propositional variables b, v, and s represent the propositions: b: Alice rode her bike today. v: Alice overslept today. s: It is sunny today. Select the logical expression that represents the statement: "Alice rode her bike today only if it was sunny today and she did not oversleep." A) b → s → ¬ v B) s ∧ ( ¬ v → b ) C) b → ( s ∧ ¬ v ) D) ( s ∧ ¬ v ) → b

C) b → ( s ∧ ¬ v )

Select the value for x that is a counter-example to the following statement: For every integer x , x < x^2 . A) x = -1/2 B) x = -1 C) x = 1 D) x = 1/2

C) x = 1

A = {a, b, c, d} X = {1, 2, 3, 4} The function f : A → X is defined as f = {(a, 4), (b, 1), (c, 4), (d, 4)} Select the set corresponding to the range of f. A) {1, 2, 3, 4} B) {1} C) {1, 4} D) ∅

C) {1, 4}

The domain for variable x is the set of all integers. Select the statement that is false. A) ∀ x ( x^2 ≥ x ) B) ∀ x ( x^2 ≠ 5 ) C) ∀ x ( x^2 > x ) D) ∃ x ( x = x )

C) ∀ x ( x^2 > x )

Prove that the statement "∀A,B,C(A∪C⊆B∪C→A⊆B)" is false.

Counterexample: Suppose A = { 2 }, B = { }, C = { 2, 3 } Then it follows: A U C = { 2, 3 } and B U C = { 2, 3 } so A U C is a subset of B U C But A is not a subset of B ( { 2 } is not a subset of { } ). Therefore, the statement above is false.

S and T are binary relations on the set {1, 2, 3, 4} and are defined by the arrow diagrams below: Select the pair that is in S∘T. A) (1, 3) B) (4, 1) C) (2, 3) D) (2, 4)

D) (2, 4)

Which statement is the contrapositive of: "If x=4, then 3x=12." A) If x = 4 , then 3 x = 12 . B) If x ≠ 4 , then 3 x ≠ 12. C) If 3 x = 12 , then x = 4 . D) If 3 x ≠ 12 , then x ≠ 4 .

D) If 3 x ≠ 12 , then x ≠ 4.

The domain of relation R is Z × Z . (a, b) is related to (c, d) if a ≤ c and b ≤ d. Which statement correctly characterizes the relation R? A) R is an equivalence relation. B) R is not an equivalence relation because R is not reflexive.term-18 C) R is not an equivalence relation because R is not transitive. D) R is not an equivalence relation because R is not symmetric.

D) R is not an equivalence relation because R is not symmetric.

Write the truth table for the proposition p→(q∨¬r).Use insert table along with symbols v, ^, ~, and -> as needed. Caution: The table menu bar will get in your way if you put your table at the very top of the text box. You can get around this by typing some extra text at the top of you text box before inserting the table. Type, "I love math," or your favorite (appropriate) one-liner, etc.

I love math p | q | r | ~r | (q∨¬r) | p→(q∨¬r) ---------------------------------- T | T | T | F | T | T | F | T | T | F | T | F | T | F | F | T | F | T | T | F | F | T | F | T | F | F | T | F | F | F | F | T |

Prove: If m and n are odd integers, then mn+3 is an even integer.

Let m and n be odd integers Then m=2k+1 and n=2j+1 for integers k and j. Now, mn+3 = (2k+1)(2j+1)+3 by substitution = 4jk + 2k +2j + 1 +3 = 4jk + 2k + 2j + 4 by algebra = 2(2jk + k + j + 2) where 2jk+k+j+2 is an integer Therefore, mn+3 is even.

The domain of a relation R is the set of integers. x is related to y under relation R if x 2 = y. Is the relation reflexive, symmetric, and/or anti-symmetric?

Reflexive: ["No"] Symmetric: [ "No"] Anti-symmetric: ["Yes"]

Give the truth assignment that shows that the argument below is not valid: p V q ¬ q ------- p <-> q Note: Enter T or F in each blank

p: ["T"] q: [ "F"]