MMW Module 5 Lesson 2 - Basta mmw PDF

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Module5: Logic Introduct Introduction ion

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everyday life, reasoning proves different points. For instance, to prove your parents that y performed well in school, you can show your grades. To prove your friends that you are a true friend to them, you just need to be a loyal and honest friend. Similarly, mathematics and computer science use mathematical logic or simply logic to prove results. To prove a theorem in mathematics, we use mathematical logic. In computer science, logic is used to prove results of computer algorithm or the correctness of a computer program. Logic is commonly referred as the science of correct reasoning, especially regarding making inferences. Mathematical reasoning and arguments are based on the rules of logic.

Learning Outcomes: At the end of this chapter, you are expected to: • • • • • • •

Determine a given sentence as statement or not. Write compound statements using the language of logic. Create truth tables of given statements. Determine the truth value of a given statement. Write the converse, inverse, and contrapositive statements of conditional statements. Determine whether statements are equivalent, tautologies or contradiction using truth tables. Appreciate the nature and concept of logic as a tool to prove results.

LESSON 2: TRUTH TABLES, EQUIVALENT STATEMENTS, AND TAUTOLOGIES Truth Table A truth table is a table used to check the “truth value” of any compound statement for all possible truth values of its simple statements. It is a way to check all possible outcomes. Presented below are the truth tables for negattion, conjunction, and disjunction for review purposes.

Truth Table with Three or More Simple Statements Compound statements that involve three or more simple statements require a standard truth table form with 2n number of rows. For instance, if the statement involves 3 simple statements, then the number of rows of truth table is 2 3 = 8. If 4 simple statements are involved, then 24 = 16 number of rows must be produced, and so on.

Lastly, use the values of (p ˅ q) and r to produce the truth values of (p ˅ q) ˄ r applying the truth value of conjunction. The shaded column is the truth table for (p ˅ q) ˄ r.

Example 9. Construct a truth table for (p ˄ q) ˄ (~r ˅ q).

Solution: Using the procedure developed above, we can produce the following table. Te shaded column is the truth table for p ˅ [~ (p ˄ ~q)]. The number below signigfies the sequence in which columns were constructed.

Equivalent Statements Two statemens are said to be equivalent if they both have the same truth value for all possible truth values of their simple statements. The symbol ≡ is used to indicates equivalents statements. Example 9. Show that ~ ( p ˅ ~ q) and ~p ˄ q are equivalent statements. Solution: Construct the truth table of the given statements and compare the results.

Since the truth values of the given statements are the same, then ~(p ˅ ~ q) ≡ ~p ˄ q.

Tautologies and Contradiction

A tautology is a statement that is “always true”. The opposite of a tautology is a contradiction, a statement that is “always false”.

If you notice in the last column, it is all “Fs”. Thus, the statement is a selfcontradiction....