The implication of a b is that: since the sun is made of gas, this makes 3 a prime number. In logic, the conditional is defined to be true unless a true hypothesis leads to a false conclusion. Note that the logical meaning of this conditional statement is not the same as its intuitive meaning. In Example 2, "The sun is made of gas" is the hypothesis and "3 is a prime number" is the conclusion. Solution: The conditional a b represents "If the sun is made of gas, then 3 is a prime number." a Then construct a truth table for this conditional. Thus, the conditional p q represents the hypothetical proposition, "If I do my homework, then I get an allowance." However, as you can see from the truth table above, doing your homework does not guarantee that you will get an allowance! In other words, there is not always a cause-and-effect relationship between the hypothesis and conclusion of a conditional statement. Solution: In Example 1, the sentence, "I do my homework" is the hypothesis and the sentence, "I get my allowance" is the conclusion. Now that we have defined a conditional, we can apply it to Example 1. Note that a conditional is a compound statement. In the truth table above, p q is only false when the hypothesis (p) is true and the conclusion (q) is false otherwise it is true. The conditional is defined to be true unless a true hypothesis leads to a false conclusion. The logical connector in a conditional statement is denoted by the symbol. Symbolized by p q, it is an if-then statement in which p is a hypothesis and q is a conclusion.
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