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CSC 301 Test 1
Name _______________________________________________________ Score____________
This is an open-book, open-note test. You MAY NOT share resource materials with anyone. You MAY NOT use calculators, computers or any electronic devices during this test. If you do not understand a question, please ask the instructor. Partial credit is possible, so show all work.
- Construct a truth-table for each of these propositions.
a. ( p ∧ q ) →( q ) b. ( p ∨ q ) →( p ∧ q )
c. ( q ) →( p ∨ q ) d. ( p ↔ q ) →( p → q )
- Which of the propositions in (1) are tautologies? Circle all that apply. a. b. c. d.
- Evaluate each of these expressions.
a. 1 1 1 0 1 ∧ (^) (1 0 0 0 1 ∨ 0 0 1 0 0)
b. 1 1 0 (^1 1) ⊕ (1 (^1 1 0 1) ⊕ 0 1 0 1 1)
- Determine the truth value of each of these statements if the domain of each variable consists of all real numbers.
a. ( 1 0 )
2
∀ x + x > b.^ ∃ x ∃ y ( x ⋅ y = 2 x + y + 1 ) {( ) }
(^2 ) c. (^) ∀ x − x = x d. (^)
∀ ∃ x y
x y x y 1
- Determine the truth value of each of the statements in (3) if the domain of each variable consists of all integers.
a. b. c. d.
- Convert the following statements into logical propositions given the indicated functions and symbolic references. Use existential and universal quantifiers as appropriate.
T ( x , y )= x isa y T(x,y) returns TRUE if the entity x is an member of the collection of all y
M = the collection of all mammals
F = the collection of all flying things
B = the collection of all bats
a. All flying mammals are bats.
b. If bats are mammals then all mammals can fly.
- Negate the following quantified expressions. Make sure that you move all negations to immediately precede predicates.
a. ∀ x^^ ∀ y {(^ x /^ y <^1 )^ ∨(^ y ≤ x )} b. ∀ x^ ∀ y ∃ z^ { x^ + y + z = x − y + z }
- Indicate whether (a) and (b) above are true when x, y, and z members of the set of real numbers.
a. b.
- Use DeMorgan's Law to evaluate each of the following expressions. In each take the negations to the lowest level possible.
a. ¬( ( A ∨ B )∨ C ) b. ¬∀ x ∀ y {( x ∧ y ) →( x ∨ y )}
