Question

You are shown a set of four cards placed on a table, each of which has a number printed on one side and a letter on the other side. The visible faces of the cards show 3, 8, A, and J. You are told that if a card shows an even number on one side, then it shows a vowel on the other side. What is the minimum number of cards you must turn over in order to be sure that this statement (or rather, open sentence) is true for all four cards?

Answer 1

Let the simple statements be p: A card has even number on one side q: A card has a vowel on one side Then the given statement is p +9 The contrapositive of the statement is - + + In word which states if a card does not show a vowel (therefore shows a con- sonant) on one of the side, then the card does not show an even number on the other side. The truth tables are as follows: p9p p + - + =p | Τ | TI FI FI Τ Ι TFF TFT F TTF T FF TTT I

So we can see the statement is false if either we find an consonant on the back of an even numbered card or find an even number on the other side of a card with consonant on one of its side. We need not to bother about the odd numbered cards or cards having vowel on one of its face because whatever such cards have on the other side does not matter as the statement would be true in all cases for such cards. So we have to turn the card with '8' and 'J' on its one side to check if they have a vowel and an odd number respectively on the other side. If either one or both of the cards fail to satisfy the condition, the state- ment is false, otherwise the statement would be true for all four cards. Therefore minimum number of cards we need to turn over is 2.