CS 381 Test
July, 2006
1. Fill in the blanks with the SHORTEST string of characters so that the
resultant proposition is valid. [20]
(a) ¬(P→Q)⇔ ¬(¬P∨Q)
⇔P∧ ¬Q
(b) (P∨(P∧Q)⇔(P∧T ) ∨(P∧Q)
⇔P∧( T ∨Q)
⇔P∧T⇔P
(c) P→(Q∧R)⇔ ¬P∨(Q∧R)
⇔ ¬P∨ ¬(¬(Q∧R))
⇔ ¬(Q∧R)→ ¬P
2. State each of the following formulas in English,if it is a wff. If it is
not a wff, then give a reason why it is not a wff. Here L(x, y) means xis
larger than y,I(x) means xis an integer and the universe is the set of real
numbers: [12]
(a) ∀x¬∃yL(x, y)
For every number, there is no number that is less than that number.
i.e. Every number is not greater than nany number.
(b) ∀x∃y¬L(x, y)
For every number there is a number that is not less than that.
i.e. Every number is not greater than some number.
(c) ∀x∃y[[I(x)∧I(y)] →L(x, y)]]
For every number x there is a number y such that if x and y are integer, then
x is gretaer than y.
i.e. For every integer, there is an integer that is less than that.
i.e. Every integer is greater than some integer.
(d) ∀x∃I(y)L(I(x), y)
This is not a proposition (well-formed formula) because an atomic formula
I(x) is an argument of another atomic formula L(x, y).
