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The truth value of a conditional statement can either be true or false. In order to show that a conditional is true, just show that every time the hypothesis is true, the conclusion is also true. To show that a conditional is false, you just need to show that every time the hypothesis is true, the conclusion is false. If you can find just 1 single example such that the hypothesis is true, but the conclusion is false, then the conditional is false. In the table below, p is the hypothesis and q is the conclusion. We denote the conditional " If p, then q" by p → q.  A few examples showing how to find the truth value of a conditional statementFind the truth value of the following conditional statements. Example #1: If a man lives in the United States of America, then the man lives in North America. The conditional is true. United States of America is a country in North America, so people who live in the United States of America live also in North America. Example #2: If the digit in the ones place of a number is 0, then the number is divisible by 10. The conditional is true. According to the divisibility rule for 10, if the last digit or the digit in the ones place of a number is 0, then the number is divisible by 10. Example #3: If you live in Alaska, then the temperature is less than 100 degrees. Alaska is usually very cold, so the first reaction may be to say that the conditional is true. However, the conditional is false. According to USA TODAY, the all-time record high temperature in Alaska is 100 degrees. This happened on June 27, 1915, in Fort Yukon. Example #4: If a number is divisible by 3, then the number is odd. The conditional is false. The number 24 is divisible by 3, but 24 is not an odd number. When the truth value of a conditional statement becomes mind-boggling.The table above states that if the hypothesis is false and the conclusion is false, then p → q is true. Take a look at the following conditional: If 3 is even, then 3 + 1 is odd. 3 is even is false. 3 + 1 is odd is also false. Why then is the conditional still true? It is true because the statement "Adding 1 to any even number will make the number odd." is a true statement. And that is the very essence of this conditional! We know 3 is not even, but suppose it is even for a second. "If 3 were even, (even for a brief second), then 3 + 1 will be odd." What is a conditional statement?
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When we previously discussed inductive reasoning we based our reasoning on examples and on data from earlier events. If we instead use facts, rules and definitions then it's called deductive reasoning. We will explain this by using an example. If you get good grades then you will get into a good college. The part after the "if": you get good grades - is called a hypotheses and the part after the "then" - you will get into a good college - is called a conclusion. Hypotheses followed by a conclusion is called an If-then statement or a conditional statement. This is noted as $$p \to q$$ This is read - if p then q. A conditional statement is false if hypothesis is true and the conclusion is false. The example above would be false if it said "if you get good grades then you will not get into a good college". If we re-arrange a conditional statement or change parts of it then we have what is called a related conditional. Example Our conditional statement is: if a population consists of 50% men then 50% of the population must be women. $$p \to q$$ If we exchange the position of the hypothesis and the conclusion we get a converse statement: if a population consists of 50% women then 50% of the population must be men. $$q\rightarrow p$$ If both statements are true or if both statements are false then the converse is true. A conditional and its converse do not mean the same thing If we negate both the hypothesis and the conclusion we get a inverse statement: if a population do not consist of 50% men then the population do not consist of 50% women. $$\sim p\rightarrow \: \sim q$$ The inverse is not true juest because the conditional is true. The inverse always has the same truth value as the converse. We could also negate a converse statement, this is called a contrapositive statement: if a population do not consist of 50% women then the population do not consist of 50% men. $$\sim q\rightarrow \: \sim p$$ The contrapositive does always have the same truth value as the conditional. If the conditional is true then the contrapositive is true. A pattern of reaoning is a true assumption if it always lead to a true conclusion. The most common patterns of reasoning are detachment and syllogism. Example If we turn of the water in the shower, then the water will stop pouring. If we call the first part p and the second part q then we know that p results in q. This means that if p is true then q will also be true. This is called the law of detachment and is noted: $$\left [ (p \to q)\wedge p \right ] \to q$$ The law of syllogism tells us that if p → q and q → r then p → r is also true. This is noted: $$\left [ (p \to q)\wedge (q \to r ) \right ] \to (p \to r)$$ Example If the following statements are true: If we turn of the water (p), then the water will stop pouring (q). If the water stops pouring (q) then we don't get wet any more (r). Then the law of syllogism tells us that if we turn of the water (p) then we don't get wet (r) must be true. Video lessonWrite a converse, inverse and contrapositive to the conditional "If you eat a whole pint of ice cream, then you won't be hungry" MAKE A REAL LIFE PROBLEM USING RATIONAL FUNCTION AND SKETCH THE GRAPH What is the answer of2/6 + 1/3 + 1/2 = 7x3(52)=4 X=-=-11-12 x= 4 pa help po need ko bukas ty po True or False 17. (3x-2y)² = 9x²-4y² 18. (2m-6)(2m-6)=4m² + 36 19. (2h-5) (2h+5) = 4h²-25 20. (x-y)(x² + xy + y²)= ׳ y³ 7 3/81 +1 1/9 = with solution A merchant sell rice at 25.00 a kilos. if how ever a buyer more than 170 kilos of rice the merchant gives a 15% discount 1. if X represent the number … 2 1/40 + 3 1/10 = with solution Let U = {0,1,2,3,4,5,6,7,8,9}; A = {1, 3, 5, 7};B = {2, 4, 6, 8} and C = {3, 6, 9, 0}. Find the following 1.h(t)=✔ײ+2×+4;h(2) |
What is the truth value of the conditional statement when the hypothesis is false and conclusion is true?
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