![]() Write the converse, inverse, contrapositive, and biconditional statements. Truth Values and Truth TablesĪnd being able to verify the truth value of conditional statements and its inverse, converse, and the contrapositive is going to be an essential part of our analysis.īut here’s a useful tip: the conditional statement and its contrapositive will always have the same truth value!Ĭonsider the implication: if n is an odd integer, then 5n+1 is even. ![]() However, the second proposition would be false since a deck of cards has two black suites: clubs and spades - therefore, we are not necessarily guaranteed that pulling a black card will result in a club. The first conditional statement is true, that if we pull a club from a deck of cards, then that card will be black. It is imperative to note that order matters when determining the validity of a statement.įor example, let’s suppose we have the proposition: “If the card is a club, then it is black,” has a very different truth value than “if the card is black, then it is a club.” Symbolic Logic Statements Converse, Inverse, and Contrapositiveįurthermore, we will learn how to take conditional statements and find new compound statements in the converse, inverse, and contrapositive form. Notice that a conditional statement “if p then q” is false when p is true and q is false, and true otherwise as noted by Northern Illinois University. Here’s a typical list of ways we can express a logical implication: What is important to note is that the arrow that separates the hypothesis from the conclusion has countless translations. “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.” “If it is sunny, then we will go to the beach.” Here are a few examples of conditional statements: In essence, it is a statement that claims that if one thing is true, then something else is true also. Let’s dive into today’s discrete lesson and find out how this works.Ī conditional statement represents an if…then statement where p is the hypothesis ( antecedent), and q is the conclusion ( consequent). In if-then form, p q means that "If you do not do your homework, then you will flunk", where p (which is equivalent to ~~ p ) is "You do not do your homework".Jenn, Founder Calcworkshop ®, 15+ Years Experience (Licensed & Certified Teacher) The given statement is "Either you do your homework or you will flunk", which is ~ p q. Let ~ p be "You do your homework" and q be "You will flunk". In expressions that include and other logical operators such as, , and ~, the order of operations is that is performed last while ~ is performed first. P is a necessary condition for q means "if not p then not q." P is a sufficient condition for q means "if p then q." It is true if both p and q have the same truth values and is false if p and q have opposite truth values. The biconditional of p and q is " p if, and only if, q" and is denoted p q. P only if q means "if not q then not p, " or equivalently, "if p then q." A conditional statement is not logically equivalent to its inverse. The inverse is "If ~ p then ~ q." Symbolically, the inverse of p q is ~p ~q. A conditional statement is not logically equivalent to its converse. The converse is "If q then p." Symbolically, the converse of p q is q p. Suppose a conditional statement of the form "If p then q" is given. A conditional statement is logically equivalent to its contrapositive. Symbolically, the contrapositive of p q is ~q ~p. The contrapositive of a conditional statement of the form "If p then q" is "If ~ q then ~ p". It is false when p is true and q is false otherwise it is true. The conditional of q by p is "If p then q" or " p implies q" and is denoted by p q. Let p and q be statement variables which apply to the following definitions. If the condition is not met, the truth of the conclusion cannot be determined the conditional statement is therefore considered to be vacuously true, or true by default. The second statement states that Sally will get the job if a certain condition (passing the exam) is met it says nothing about what will happen if the condition is not met. The hypothesis in the first statement is "144 is divisible by 12", and the conclusion is "144 is divisible by 3". Let q stand for the statements "Sally will get the job" and "144 is divisible by 3". Let p stand for the statements "Sally passes the exam" and "144 is divisible by 12". ![]() If 144 is divisible by 12, 144 is divisible by 3. If Sally passes the exam, then she will get the job. ![]() For instance, consider the two following statements: ![]() In conditional statements, "If p then q" is denoted symbolically by " p q" p is called the hypothesis and q is called the conclusion. Negation, Converse & Inverse | Truth Table For Conditional Statemen ts Conditional Statement Forms Conditional Statements | Definitions | Representation of If-Then as Or ![]()
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