Final-Module-7-Statements-Related-to-Conditional-Statements-and-Logical-Equivalence.pdf

BULACAN STATE UNIVERSITY COLLEGE OF SCIENCE MMW 101 MATHEMATICS IN THE MODERN WORLD Module 7 Statements Related to Conditional Statements and Logical Equivalence “Achieving Universal Understanding and Peace Through the Language of Math...

BULACAN STATE UNIVERSITY COLLEGE OF SCIENCE MMW 101 MATHEMATICS IN THE MODERN WORLD Module 7 Statements Related to Conditional Statements and Logical Equivalence “Achieving Universal Understanding and Peace Through the Language of Mathematics”  62 Statements Related to Conditional Statements and Logical Equivalence Objectives of the Module At the end of the module, you should be able to: 1. determine whether propositions are logically equivalent, and 2. state the converse, inverse, and contrapositive of conditional statements. Logical Equivalence Two statements having the same truth values in all possible cases are logically equivalent. Symbolic form: p q or p ≡ q (read as p and q are logically equivalent) Examples: 1. Show that p → q and ~p ∨ q are logically equivalent. Solution: Step 1: Begin with the standard truth table form. Step 2: Negate p and then write the results on a new column. Step 3: Write the truth values of p → q in the next column. Step 4: Using the truth values of the negation of p (in step 2) and q (in column 2), write the truth values of ~p ∨ q in the last column. p q ~p p→q ~p ∨ q T T F T T T F F F F F T T T T F F T T T Since p → q and ~p ∨ q have the same truth values in all possible cases, they are logically equivalent. In symbolic form: p → q ⇔ ~p ∨ q or p → q ≡ ~p ∨ q.  63 2. Is ~p ∧ ~q logically equivalent to p ∨ q? Let us examine the truth table below. p q ~p ~q ~p ∧ ~q p∨q T T F F F T T F F T F T F T T F F T F F T T T F Since the truth values of ~p ∧ ~q in all cases are not the same as the truth values of p ∨ q, then ~p ∧ ~q is not logically equivalent to p ∨ q or in symbols, ~p ∧ ~ q ⇎ p ∨ q. 3. Verify if ~(p → q) is logically equivalent to p ∧ ~q. p q ~q p →q ~ (p → q) p ∧ ~q T T F T F F T F T F T T F T F T F F F F T T F F From the truth table, we can see that ~ (p → q) have the same truth values as p ∧ ~ q. Therefore, they are logically equivalent. Try this! Is q ∧ ~p logically equivalent to ~p ∨ q? Use the truth table to show your answer.  64 The Converse, the Inverse, and the Contrapositive There are three statements related to a conditional statement. These are the converse, the inverse, and the contrapositive. Given: conditional statement p → q Converse q→p Interchange the hypothesis (p) and the conclusion (q). Inverse ~p → ~q Negate both the hypothesis (p) and the conclusion (q). Contrapositive ~q → ~p Interchange the negated hypothesis (p) and the negated conclusion (q). Examples: Write the converse, the inverse, and the contrapositive of the following conditional statements: 1. If I get the loan, then I will buy a new motorbike. 2. If you are smart, then you can get the job. Solution: 1. If I get the loan, then I will buy a new motorbike. Converse: If I will buy a new motorbike, then I get the loan. Inverse: If I do not get the loan, then I will not buy a new motorbike. Contrapositive: If I will not buy a new motorbike, then I do not get the loan. 2. If you are smart, then you can get the job. Converse: If you can get the job, then you are smart. Inverse: If you are not smart, then you cannot get the job. Contrapositive: If you cannot get the job, then you are not smart. Try this! Tell the converse, the inverse, and the contrapositive of the conditional statement, "I feel nauseous whenever I stay up late at night."  65 Truth Table for the Conditional and its Related Statements The truth table for the conditional and its related statements is shown below. Conditional Converse Inverse Contrapositive p q ~p ~q p→q q→p ~p → ~q ~q → ~p T T F F T T T T T F F T F T T F F T T F T F F T F F T T T T T T The table also shows that any conditional statement is logically equivalent to its contrapositive, and its converse is logically equivalent to its inverse. Notation: p → q ≡ ~q → ~p q → p ≡ ~p → ~q  66 References Aufmann, R.N., et. Al. (2018). Mathematics in the Modern World (14th ed.). Sampaloc, Manila: Rex Book Store, Inc. Baltazar, E., Ragasa, C., & Evangelista, J. (2018). Mathematics in the Modern World. Quezon City: C&E Publishing, Inc. Earnheart, R. and Adina, E. (2018). Math in the Modern World. Quezon City : C &E Publishing, Inc. Malang, P., Malang, B., & Tiongson, I. (2011). Discrete Structure. San Rafael, Bulacan : HFM Publishing. Rosen, K.H. (1988). Discrete Mathematics and Its Applications. New York : The Random House. Simpson, A. (2002). Discrete Mathematics by Example. United Kingdom : McGraw- Hill Education.