1. www.company.com Module Code MA0003NI: Computing mathematics Lecture for Week 8 2012-2013Autumn

2. www.company.com Agenda Week 8 Lecture coverage: –Another Boolean Algebra, –Circuit Design –The Bi-conditional of Two Propositions – Logical Consequence

3. www.company.com Another Boolean Algebra Review: A high pulse (1) with T and a low pulse (0) with F the truth tables A + B, A · B and Ā for the OR, AND and NOT gates are identical to those for p ∨ q, p ∧ q and ¬p.

4. www.company.com Another Boolean Algebra (Cont.)

5. www.company.com Another Boolean Algebra (Cont.)

6. www.company.com Another Boolean Algebra (Cont.)

7. www.company.com Another Boolean Algebra (Cont.)

8. www.company.com Circuit Design Consider the circuit below: This circuit uses three gates (two NOT gates and an AND gate).

9. www.company.com Circuit Design (Cont.) However, by de Morgan’s laws = Now the circuit with the output is This circuit uses only two gates (one NOT gates and an OR gate). It is equivalent to the original circuit and is easier to build.

10. www.company.com Circuit Design (Cont.) Note we can simplify the expression algebraically (using any of the laws 1 – 11) our simplified expression will represent a circuit which is equivalent to the original one. The original circuit can actually be replaced by the new one and will produce identical output for all possible combinations of input values (i.e. pulses). The goal in the simplification is to reduce the number of gates used by the circuit.

11. www.company.com Circuit Design Three Steps starting from given circuit requirements in the form of a table. 1.Formulate a Boolean expression for the output function from the given table. 2.Simplify this expression as much as possible using Boolean algebra. 3.Draw the circuit corresponding to the simplified output function.

12. www.company.com Circuit Design (Cont.) Example: We will design a circuit corresponding to the following truth table. The output function is labelled X.

13. www.company.com Circuit Design (Cont.) Step 1. First scan the output column for occurrences of 1. In this example there are three (lines 1, 2 and 4).

14. www.company.com Circuit Design (Cont.)

15. www.company.com Circuit Design (Cont.)

16. www.company.com Circuit Design (Cont.)

17. www.company.com Circuit Design (Cont.) Another Example: Design a Circuit corresponding to:

18. www.company.com Circuit Design (Cont.)

19. www.company.com Circuit Design (Cont.)

20. www.company.com Simplify and construct the logic circuit 1. A’.B’ + (A.B)’ 2. (A + B).(A + B) + A.(A + B’) 3. (A’. B + A.B’)’ 4. ((A + C).(AB)’ + (BC + A’)’)’

21. www.company.com The inverse of p → q For any two propositions p and q the conditional  p →  q is called the inverse of p → q. The truth table for  p →  q is shown below together with that for p → q for comparison.. pq pp  q p→q p→qp→q TTFFTT TFFTFT FTTFTF FFTTTT

22. www.company.com The converse of p → q For any two propositions p and q the conditional q → p is called the converse of p → q. The truth table for q → p is shown below together with that for p → q for comparison. Columns 2 and 3 of the two tables differ so q → p ≢ p → q. This means, of course, that the connective → is not commutative.

23. www.company.com The contrapositive of p → q Consider the following example: The truth table for q → p is shown below together with that for p → q for comparison. Consequently ¬q → ¬p ≡ p → q. The proposition ¬q → ¬p is called the contrapositive of p → q. It is equivalent to p → q. (example) (Commutative Law) (double negation Law) (example)

24. www.company.com The Contrapositive of p → q For any two propositions p and q the conditional  q →  p is called the contrapositive of p → q. The truth table for  q →  p is shown below together with that for p → q for comparison.. pq pp  q p→q q→pq→p TTFFTT TFFTFF FTTFTT FFTTTT

25. www.company.com The Bi-conditional of Two Propositions If propositions p and q are combined in the form “p if and only if q” the resulting proposition is called the bi-conditional of p and q, or simply the bi-conditional. The logical connective is the “if and only if ” connective with symbol ↔. The proposition p ↔ q is really shorthand for (p → q) ∧ (q → p)

26. www.company.com The Bi-conditional of Two Propositions (Cont,) We can construct its truth table as below We conclude that the truth table for p ↔ q is note that p ↔ q means p and q are true and false together

27. www.company.com The Bi-conditional of Two Propositions (Cont,)

28. www.company.com Logical Consequence

29. www.company.com Logical Consequence (Cont.) Consider the following example:

30. www.company.com Logical Consequence (Cont.)

31. www.company.com ARGUMENTS An argument is a relationship between a set of propositions,, called premises, and another proposition Q, called the conclusion. An argument is denoted by ├ Q An argument ├ Q is said to be valid iff is a tautology.

32. www.company.com Questions Determine the validity of the following arguments: 1. p v q, ~p Ⱶ q 2. p → q, q → r, ~r Ⱶ ~p 3. If you do not study you will fail your examination. You failed therefore you did not study. 4. If I am not in Malaysia, then I am not happy; if I am happy, then I am singing; I am into singing; therefore, I am not in Malaysia