![]() ![]() As you develop your mathematical intuition for ideas like these, you will feel more and more comfortable with the sometimes surprising results. Finally, the negation of a statement may not always be what you expect – for example here we saw that the negation of the conditional is actually an “and” statement. When trying to understand logical statements and how to negate them, it can be helpful to consider equivalent statements and to utilize truth tables to check your work at every stage. Let’s try it for this negation.Īs you can see, we end up with the same truth values for each statement, so they are equivalent and we have verified that we did the negation correctly. (p and not q) Verifying with a truth tableĪlthough the work above is enough, you can always double check your results using a truth table. Negation: I run fast and I do not get tired. Statement: If I run fast, then I get tired. If we were to apply this to a real-life statement, then we would have something like the following. If 144 is divisible by 12, 144 is divisible by 3. For instance, consider the two following statements: If Sally passes the exam, then she will get the job. This shows that the negation of “p implies q” is “p and not q”. In conditional statements, 'If p then q ' is denoted symbolically by ' p q ' p is called the hypothesis and q is called the conclusion. We will see how these statements work with an example. The inverse of the conditional statement is If not P then not Q. The contrapositive of the conditional statement is If not Q then not P. Now we can use De Morgan’s laws to negate this statement: The converse of the conditional statement is If Q then P. This is one of those things you might have to think about a bit for it to make sense, but even with that, the truth table shows that the two statements are equivalent. Thus, “p implies q” is equivalent to “q or not p”, which is typically written as “not p or q”. Or, we could have “not p”, and therefore, we would not have q (so we could use possibility 2 as not p). The law of detachment has a prescribed pattern. Now, let's get back to the pattern alluded to earlier. We could have “p”, and therefore “q” (so q is possibility 1). The conditional statement can now be rewritten with the symbols as: If p, then q. Studying for the test is a sufficient condition for passing the class. If the sky is clear, then we will be able to see the stars. Why is this true? Given “p implies q”, there are two possibilities. Here are a few examples of conditional statements: If it is sunny, then we will go to the beach. Consider this as you review the following truth table. Thus, if you know p, then the logical conclusion is q. One way to write the conditional is: “if p, then q”. Inverse, & Contrapositive Conditional & Biconditional Statements, Logic, Geometry. Let’s get started with an important equivalent statement to the conditional. The table below shows the logic operators and their symbols. But, if we use an equivalent logical statement, some rules like De Morgan’s laws, and a truth table to double-check everything, then it isn’t quite so difficult to figure out. (e) \(r\Rightarrow p\), which is true regardless of the whether \(r\) is true or false.The negation of the conditional statement “p implies q” can be a little confusing to think about. (d) \(q\Rightarrow r\), which is true regardless of the whether \(r\) is true or false. (c) \((p\vee q)\Rightarrow r\), which is true if \(r\) is true, and is false if \(r\) is false. (b) \(p\Rightarrow r\), which is true if \(r\) is true, and is false if \(r\) is false. For Niagara Falls to be in New York, it is sufficient that New York City will have more than 40 inches of snow in 2525.For New York City to be the state capital of New York, it is necessary that New York City will have more than 40 inches of snow in 2525.e.Niagara Falls is in New York or New York City is the state capital of New York implies that New York City will have more than 40 inches of snow in 2525.Niagara Falls is in New York only if New York City will have more than 40 inches of snow in 2525.If Niagara Falls is in New York, then New York City is the state capital of New York.What is their truth value if \(r\) is true? What if \(r\) is false? Represent each of the following statements symbolically. ![]() The statement \(p\) is true, and the statement \(q\) is false. New York City will have more than 40 inches of snow in 2525. for alogic NAND gate is denoted by a single dot or full stop symbol, (. The contrapositive of a conditional statement of p q is :q :p. Section 5.4: The Conditional and Related StatementsEquivalent Forms of the. p is a su cient condition for q and q is a necessary condition for p. New York City is the state capital of New York. :(pq) :p:q:(pq) :p:q If p and q are propositions, the conditional if p then q' (or p only if q' or q if p), denoted by p q, is false when p is true and q is false otherwise it is true. \)Ĭonsider the following statements: \(p\): ![]()
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