The note for Exercise 10 also applies to this exercise. The two statements in this activity are logically equivalent. We now have the choice of proving either of these statements. If we prove one, we prove the other, or if we show one is false, the other is also false. The second statement is Theorem 1. Basically, this means these statements are equivalent, and we make the following definition: Definition Two expressions are logically equivalent provided that they have the same truth value for all possible combinations of truth values for all variables appearing in the two expressions.
Proof The first equivalency in Theorem 2. So what does it mean to say that the conditional statement If you do not clean your room, then you cannot watch TV, is false? So the negation of this can be written as You do not clean your room and you can watch TV. Another Method of Establishing Logical Equivalencies We have seen that it often possible to use a truth table to establish a logical equivalency. Progress Check 2.
Let a and b be integers. Answer Add texts here. Do not delete this text first. Exercises for Section 2. That sounds like a mouthful, but what it means is that "not A and B " is logically equivalent to "not A or not B". Similarly, the negation of a disjunction of 2 statements is logically equivalent to the conjunction of each statement's negation.
Put simply, "not A or B " is logically equivalent to "not A and not B". Symbolically, this can be written as. These two statements are logically equivalent click here for definition , and this can be verified by using a truth table. Table of Logical Equivalencies:. The following table can be used to help reduce compound statements to simpler forms.
Given statement variables p, q, and r , a tautology t and a contradiction c, the following rules of logic hold:. Logical equivalencies can be used to simplify statement forms, to confirm or disprove an equivalency, to create efficient and logically correct computer programs, or to aid in the design of digital logic circuits.
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I have given the answer to you, because it explains it more. I just worry about them being sentences here. Add a comment. For example, the contrapositive of "If it is raining then the grass is wet" is "If the grass is not wet then it is not raining. A conditional statement is logically equivalent to its contrapositive. Converse: Suppose a conditional statement of the form "If p then q" is given. The converse is "If q then p. A conditional statement is not logically equivalent to its converse.
Equivalent Statements are statements that are written differently, but hold the same logical equivalence. Definition: An argument consists of a sequence of statements called premises and a statement called a conclusion.
Now: Rewrite this argument in its general form by defining appro- priate propositional variables. This is one example of an argument form that is called disjunctive syllogism. Discrete Mathematics - Propositional Logic. The rules of mathematical logic specify methods of reasoning mathematical statements. Compound sentences of the form "P if and only if Q" are true when P and Q are both false or are both true; this compound sentence is false otherwise.
It says that P and Q have the same truth values; when "P if and only if Q" is true, it is often said that P and Q are logically equivalent. Concepts and notations from discrete mathematics are useful in studying and describing objects and problems in branches of computer science, such as computer algorithms, programming languages, cryptography, automated theorem proving, and software development.
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