Conditional Statement Definition: A Conditional Statement is... symbolized by p https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition al.gif q, it is an if-then statement in which p is a hypothesis and q is a conclusion. The logical connector in a conditional statement is denoted by the symbol https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition al.gif . The conditional is defined to be true unless a true hypothesis leads to a false conclusion. A truth table for p https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition al.gif q is shown below. p q p https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition al.gif q T T T T F F F T T F F T In the truth table above, p https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition al.gif q is only false when the hypothesis (p) is true and the conclusion (q) is false; otherwise it is true. Note that a conditional is a compound statement. Now that we have defined a conditional, we can apply it to Example 1. Example 1: Given: p: I do my homework. q: I get my allowance. Problem: What does p https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition al.gif q represent? Solution: In Example 1, the sentence, "I do my homework" is the hypothesis and the sentence, "I get my allowance" is the conclusion. Thus, the conditional p https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition al.gif 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. Example 2: Given: a: The sun is made of gas. b: 3 is a prime number. Problem: Write a https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition al.gif b as a sentence. Then construct a truth table for this conditional. Solution: The conditional a https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition al.gif b represents "If the sun is made of gas, then 3 is a prime number." a b a https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition al.gif b T T T T F F F T T F F T In Example 2, "The sun is made of gas" is the hypothesis and "3 is a prime number" is the conclusion. Note that the logical meaning of this conditional statement is not the same as its intuitive meaning. In logic, the conditional is defined to be true unless a true hypothesis leads to a false conclusion. The implication of a https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition al.gif b is that: since the sun is made of gas, this makes 3 a prime number. However, intuitively, we know that this is false because the sun and the number three have nothing to do with one another! Therefore, the logical conditional allows implications to be true even when the hypothesis and the conclusion have no logical connection. Example 3: Given: x: Gisele has a math assignment. y: David owns a car. Problem: Write x https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition al.gif y as a sentence. Solution: The conditional x https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition al.gif y represents, "If Gisele has a math assignment, then David owns a car. In the following examples, we are given the truth values of the hypothesis and the conclusion and asked to determine the truth value of the conditional. Example 4: Given: r: 8 is an odd number. false s: 9 is composite. true Problem: What is the truth value of r https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition al.gif s? Solution: Since hypothesis r is false and conclusion s is true, the conditional r https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition al.gif s is true. Example 5: Given: r: 8 is an odd number. false s: 9 is composite. true Problem: What is the truth value of s https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition al.gif r? Solution: Since hypothesis s is true and conclusion r is false, the conditional s https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition al.gif r is false. Example 6: Given: p: 7^2 = 49. true q: A rectangle does not have 4 sides. false r: Harrison Ford is an American actor. true s: A square is not a quadrilateral. false Problem: Write each conditional below as a sentence. Then indicate its truth value. 1. p https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition al.gif q If 7^2 is equal to 49, then a rectangle does not have 4 sides. false 2. q https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition al.gif r If a rectangle does not have 4 sides, then Harrison Ford is an American actor. true 3. p https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition al.gif r If 7^2 is equal to 49, then Harrison Ford is an American actor. true 4. q https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition al.gif s If a rectangle does not have 4 sides, then a square is not a quadrilateral. true 5. r https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition al.gif ~p If Harrison Ford is an American actor, then 7^2 is not equal to 49. false 6. ~r https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition al.gif p If Harrison Ford is not an American actor, then 7^2 is equal to 49. true Note that in item 5, the conclusion is the negation of p. Also, in item 6, the hypothesis is the negation of r. ___________________________________________________________________________________________________ Summary: A conditional statement, symbolized by p https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition al.gif q, is an if-then statement in which p is a hypothesis and q is a conclusion. The conditional is defined to be true unless a true hypothesis leads to a false conclusion. ___________________________________________________________________________________________________ Exercises Directions: Read each question below. Select your answer by clicking on its button. Feedback to your answer is provided in the RESULTS BOX. If you make a mistake, choose a different button. 1. Which of the following is a conditional statement? Začátek formuláře (_) Amy plays soccer or Bill plays hockey. (_) Bill plays hockey when Amy plays soccer. (_) If Amy plays soccer then Bill plays hockey. (_) None of the above. RESULTS BOX: _____________________________________________ Konec formuláře 2. Given: r: You give me twenty dollars. s: I will be your best friend. Problem: Which of the following statements represents, "If you give me twenty dollars, then I will be your best friend"? Začátek formuláře (_) r https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/and.gif s (_) r https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition al_transp.gif s (_) s https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition al_transp.gif r (_) None of the above. RESULTS BOX: _____________________________________________ Konec formuláře 3. What is the truth value of r https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition al_transp.gif s when the hypothesis is false and the conclusion is true in Example 2? Začátek formuláře (_) True (_) False (_) Not enough information was given. (_) None of the above. RESULTS BOX: _____________________________________________ Konec formuláře 4. Given: a: x is prime. b: x is odd. Problem: What is the truth value of a https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition al_transp.gif b when x = 2? Začátek formuláře (_) True (_) False (_) Not enough information was given. (_) None of the above. RESULTS BOX: _____________________________________________ Konec formuláře 5. What is the truth value of a https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition al_transp.gif b when x = 9 in Exercise 4? Začátek formuláře (_) True (_) False (_) Not enough information was given. (_) None of the above. RESULTS BOX: _____________________________________________ Konec formuláře