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Consider the conditional statement βIf π΄, then π΅,β where the hypothesis π΄ is βπ₯ and π¦ are even numbersβ and the conclusion π΅ is βπ₯ plus π¦ is even.β
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Complete the table to give the truth value of the conditional statement and its converse, inverse, and contrapositive.
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One way to test truth value is to see if you can give a counterexample, is to see if you can give a place where that statement is not true.
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If we find even one counterexample, the truth value of the statement is false.
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Weβll start here, if π΄, then π΅.
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π₯ and π¦ are even.
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Then, π₯ plus π¦ is even.
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Two plus two equals four is an example.
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Is there any time that two even numbers would add up to an odd number?
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Four plus four equals eight.
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10 plus 10 equals 20.
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When we add two even numbers, we always get an even number.
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The truth value for βif π΄, then π΅β is true.
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Now weβre moving on, if π΅, then π΄.
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π₯ plus π¦ is even.
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Therefore, π₯ and π¦ are even numbers.
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Can you think of any example where π₯ plus π¦ is an even number but the two values that you add together are not even?
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For example, six is an even number.
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We can add three plus three together to equal six.
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In this case, the first statement is true.
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π₯ plus π¦ is even.
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But the second statement is not true.
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Both of the add-ins are not even.
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And that makes the converse, if π΅, then π΄, false.
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Next one, if not π΄, then not π΅.
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If not π΄, that means π₯ and π¦ are not even.
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Then, π₯ plus π¦ is not even.
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Weβre looking for any counterexamples, any places where this would not be true.
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π₯ and π¦ are not even.
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Letβs start there.
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We have three plus five.
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So three is not even.
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Five is not even.
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Then, π₯ plus π¦ equals eight.
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But eight is an even number.
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We have a case where π₯ and π¦ are not even.
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But that does not mean that π₯ plus π¦ is not even because it is.
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Thereβs a counterexample, which means the truth value of that statement is false.
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The inverse of if π΄, then π΅ is false.
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Our last statement, the contrapositive, if not π΅, then not π΄.
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Not π΅ would mean π₯ plus π¦ is not even.
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Not π΄, and that means π₯ plus π¦ are not even.
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So we need a not even value for π₯ plus π¦.
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Weβll say that π₯ plus π¦ equals nine.
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Not π΄ means that π₯ and π¦ are not even.
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We could say one plus eight.
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And one of those values, eight, is even.
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But both of them are not.
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π₯ and π¦ are not even.
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That means not π΅ is true and not π΄ is also true so far.
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And hereβs the question here.
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Can two even numbers ever add up to an odd number?
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No.
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Since two even numbers can never add up to an odd number, the contrapositive of this statement is true, if not π΅, then not π΄.
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Our table is complete with true, false, false, true.