By Earl E Swartzlander
The publication presents the various easy papers in machine mathematics. those papers describe the suggestions and uncomplicated operations (in the phrases of the unique builders) that may be helpful to the designers of pcs and embedded structures. even supposing the main target is at the easy operations of addition, multiplication and department, complex suggestions comparable to logarithmic mathematics and the calculations of effortless services also are coated.
Readership: Graduate scholars and study pros drawn to machine mathematics.
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Extra info for Computer Arithmetic: Volume I
The substitution rule and the replacement laws give us a means of establishing logical equivalences between propositions without drawing up a truth table. We demonstrate this in the following example. 9 Prove that (¯ p ∧ q) ∨ (p ∨ q) ≡ p¯. Solution (¯ p ∧ q) ∨ (p ∨ q) ≡ (¯ p ∧ q) ∨ (¯ p ∧ q¯) ≡ p¯ ∧ (q ∨ q¯) ≡ p¯ ∧ t ≡ p¯. 4 1. 9. (i) (ii) (iii) (iv) (v) 2. (p ∧ p) ∨ (¯ p ∨ p¯) ≡ t. (p ∧ q) ∧ q ≡ p ∧ q. p → q ≡ p ∧ q¯. (p ∧ q) → r ≡ (¯ p ∨ q¯) ∨ r. q ∧ [(p ∨ q) ∧ (¯ q ∧ p¯)] ≡ q ∧ (q ∨ p). 9 to show that p ∧ (q ∨ p¯) is logically equivalent to p ∧ q.
Converse: q → p: If Sara will sing then Jack plays his guitar. Inverse: p¯ → q¯: If Jack doesn’t play his guitar then Sara won’t sing. Contrapositive: q¯ → p¯: If Sara won’t sing then Jack doesn’t play his guitar. As we have shown, ‘If Jack plays his guitar then Sara will sing’ and ‘If Sara won’t sing then Jack doesn’t play his guitar’ are equivalent propositions. 3 1. Prove that (p → q) ≡ (¯ p ∨ q). 2. Prove that (p ∧ q) and (p → q¯) are logically equivalent propositions. 3. Prove that (p 4. Prove that p logically implies (q → p).
10. Consider a new connective, denoted by |, where p|q is defined by the following truth table: p q p|q T T F T F T F T T F F T Show that: (i) (ii) p¯ ≡ (p|p) (p ∧ q) ≡ (p|q)|(p|q). 9 above to deduce that a proposition involving any of the five familiar connectives can be written in a logically equivalent form which uses only the connective denoted by |. 11. State the converse, inverse and contrapositive of the proposition: ‘If it’s not Sunday then the supermarket is open until midnight’. 4. ) These are often referred to as ‘replacement laws’ because, as we shall see later, there are situations where it is useful to substitute one proposition for another logically equivalent form.