Here the conditional statement logic is, if not B, then not A (~B → ~A) Biconditional Statement When the hypothesis and conclusion are negative and simultaneously interchanged, then the statement is contrapositive.Ĭontrapositive: “If yesterday was not Sunday, then today is not Monday” Here the conditional statement logic is, If not A, then not B (~A → ~B) Contrapositive Statement Inverse: “If today is not Monday, then yesterday was not Sunday.” When both the hypothesis and conclusion of the conditional statement are negative, it is termed as an inverse of the statement.Ĭonditional Statement:“If today is Monday, then yesterday was Sunday”. Here the conditional statement logic is, If B, then A (B → A) Inverse of Statement When hypothesis and conclusion are switched or interchanged, it is termed as converse statement.Ĭonditional Statement: “If today is Monday, then yesterday was Sunday.”Ĭonverse: “If yesterday was Sunday, then today is Monday.” Let us consider hypothesis as statement A and Conclusion as statement B. To check whether the statement is true or false here, we have subsequent parts of a conditional statement. On interchanging the form of statement the relationship gets changed. Let us consider the above-stated example to understand the parts of a conditional statement.Ĭonditional Statement: If today is Monday, then yesterday was Sunday. Hypothesis (if) and Conclusion (then) are the two main parts that form a conditional statement. What Are the Parts of a Conditional Statement? Here are two more conditional statement examplesĮxample 1: If a number is divisible by 4, then it is divisible by 2.Įxample 2: If today is Monday, then yesterday was Sunday. '\(\rightarrow\)' is the symbol used to represent the relation between two statements. For example, A\(\rightarrow\)B. Here 'p' refers to 'hypothesis' and 'q' refers to 'conclusion'.įor example, "If Cliff is thirsty, then she drinks water." (In Table 1.1, T stands for “true” and F stands for “false.What Is Meant By a Conditional Statement?Ī statement that is of the form "If p, then q" is a conditional statement. This is summarized in Table 1.1, which is called a truth table for the conditional statement \(P \to Q\). Using this as a guide, we define the conditional statement \(P \to Q\) to be false only when \(P\) is true and \(Q\) is false, that is, only when the hypothesis is true and the conclusion is false. It says nothing about the truth value of \(Q\) when \(P\) is false. The conditional statement \(P \to Q\) means that \(Q\) is true whenever \(P\) is true. Because conditional statements are used so often, a symbolic shorthand notation is used to represent the conditional statement “If \(P\) then \(Q\).” We will use the notation \(P \to Q\) to represent “If \(P\) then \(Q\).” When \(P\) and \(Q\) are statements, it seems reasonable that the truth value (true or false) of the conditional statement \(P \to Q\) depends on the truth values of \(P\) and \(Q\). Intuitively, “If \(P\) then \(Q\)” means that \(Q\) must be true whenever \(P\) is true. For this conditional statement, \(P\) is called the hypothesis and \(Q\) is called the conclusion. Thinking out loud is often a useful brainstorming method that helps generate new ideas.Ī conditional statement is a statement that can be written in the form “If \(P\) then \(Q\),” where \(P\) and \(Q\) are sentences. When we work with someone else, we can compare notes and articulate our ideas. Working together is often more fruitful than working alone. We could use this identity to argue that the conjecture “for all real numbers \(x\), \(\sin (2x) = 2 \sin(x)\)” is false, but if we do, it is still a good idea to give a specific counterexample as we did before.We should recall (or find) thatįor all real numbers \(x\), \ For the conjecture that \(\sin (2x) = 2 \sin(x)\), for all real numbers \(x\), we might recall that there are trigonometric identities called “double angle identities.” We may even remember the correct identity for \(\sin (2x)\), but if we do not, we can always look it up. We must make use of our acquired mathematical knowledge. We cannot start from square one every time we explore a statement. (We will prove that this statement is true in the next section.) However, it is not possible to test every pair of odd integers, and so we can only say that the conjecture appears to be true. We can do lots of calculation, such as \(3 + 7 = 10\) and \(5 + 11 = 16\), and find that every time we add two odd integers, the sum is an even integer. If \(x\) and \(y\) are odd integers, then \(x + y\) is an even integer. As an example, consider the conjecture that The best we can say is that our examples indicate the conjecture is true. However, if we do not find a counterexample for a conjecture, we usually cannot claim the conjecture is true. Since \(1 \ne \sqrt2\), these calculations show that this conjecture is false.
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