By Harris Kwong
This can be a textual content that covers the normal subject matters in a sophomore-level direction in discrete arithmetic: good judgment, units, evidence suggestions, easy quantity conception, features, kinfolk, and common combinatorics, with an emphasis on motivation. It explains and clarifies the unwritten conventions in arithmetic, and courses the scholars via a close dialogue on how an explanation is revised from its draft to a last polished shape. Hands-on routines aid scholars comprehend an idea quickly after studying it. The textual content adopts a spiral method: many subject matters are revisited a number of occasions, occasionally from a unique point of view or at a better point of complexity. The aim is to slowly enhance scholars’ problem-solving and writing talents.
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7. Construct the truth tables for the following expressions: (a) (p ∧ q) ∨ r (b) (p ∨ q) ⇒ (p ∧ r) Hint: To help you get started, fill in the blanks. (a) p T T T T F F F F r p ∧ q (p ∧ q) ∨ r T F T F T F T F q T T F F T T F F (b) p T T T T F F F F q T T F F T T F F r T F T F T F T F p∨q p∧r (p ∨ q) ⇒ (p ∧ r) 8. Construct the truth tables for the following expressions: (a) (p ⇒ q) ∨ (p ⇒ q) (b) (p ⇒ q) ∧ (p ⇒ q) 9. Determine (you may use a truth table) the truth value of p if (a) (p ∧ q) ⇒ (q ∨ r) is false (b) (q ∧ r) ⇒ (p ∧ q) is false 10.
X2 ≤ 0, then x ≤ 0. x2 > 0, then x > 0. 36 Chapter 2 Logic 12. Determine whether the following formulas are tautologies, contradictions, or neither: (a) (p ⇒ q) ∧ p (b) (p ⇒ q) ∧ (p ∧ q) (c) (p ⇒ q) ∧ q 13. Simplify the following formulas: (a) p ∧ (p ∧ q) (b) p ∨ q (c) p ⇒ q 14. 6 (b) p ∧ q (c) p ∧ (p ∨ q) Logical Quantifiers The expression x>5 is neither true nor false. In fact, we cannot even determine its truth value unless we know the value of x. This is an example of a propositional function, because it behaves like a function of x, it becomes a proposition when a specific value is assigned to x.
X2 − 4x + 3 − 0 ⇔ x = 3. x2 > y 2 ⇔ x > y. 6. Determine the truth values of the following statements (assuming that x and y are real numbers): (a) (b) (c) (d) u is a vowel if and only if b is a consonant. x2 + y 2 = 0 if and only if x = 0 and y = 0. x2 − 4x + 4 = 0 if and only if x = 2. xy = 0 if and only if x and y are both positive. 7. We have seen that a number n is even if and only if n = 2q for some integer q. Accordingly, what can you say about an odd number? 8. We also say that an integer n is even if it is divisible by 2, hence it can be written as n = 2q for some integer q, where q represents the quotient when n is divided by 2.