Math Propositions and Their Combinations Fall 2003 Page 1. Propositions

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1 Math Propositions and Their Combinations Fall 2003 Page 1 Propositions One way that mathematical language differs from ordinary everyday English (OEE) is in what counts as a statement. OEE accepts a wide variety of statements: Questions Why me? When is lunch? What is the air speed velocity of a swallow? Exclamations, commands Dag, yo! Show me the money! Expressions that are vague or ambiguous Metaphors Statements that are open to debate or are a matter of opinion. Statements whose truth or falsity depend on context It was the best of times, it was the worst of times. People are strange. My love is like a red, red, rose. John is like a brother to me. Mathematics is the best major in the world. Taking colloidal silver will boost your immune system. Strongbad is awesome. People are dishonest. Mice eat cheese. Conservatives are pro-life. In the language of formal mathematics (LFM), we want to deal only with statements that can clearly be completely true or completely false, with no gray area and no dependence on context. We call such statements propositions.

2 Math Propositions and Their Combinations Fall 2003 Page 2 Examples of Propositions: 6-3i is a complex number 18 is a prime number Your teacher in Math 190 Fall 2003 was born on a Tuesday. The Gross National Product of Japan is greater than that of the United States. A square has exactly 3 right angles. Every compact Hausdorff space is normal. The order of a subgroup of any finite group divides the order of the group.

3 Math Propositions and Their Combinations Fall 2003 Page 3 Combining Propositions In OEE, we have lots of ways of combining statements together: I tried to add the class but the computer wouldn t let me. I stood in line for 45 minutes, and found out I was in the wrong line. I m either going to take Conversational Norwegian or Intro to Art History. If I can t get into Chem 101, [then] it will screw up my whole schedule. If they don t turn that radio down, [then] I m gonna call the police. In the language of formal mathematics (LFM), we also have ways of combining propositions using words like and, or, if then, and so forth. BUT they often don t have exactly the same meanings as they do in OEE.

4 Math Propositions and Their Combinations Fall 2003 Page 4 Example: OR In OEE, when we say A or B we usually mean either A, or B, but not both. Examples: You need to fish or cut bait. (Sink or swim, win or lose) Stop that or I ll punch you. Mom or I will take you to the mall. We will go to the movie or the game. Either she ll say yes or she ll say no; but at least you ll know. Should I be a chartered accountant, or a lumberjack? You must take either Math 315 or Math 371 Do you want soup or salad? In some of these cases, you would be surprised and maybe angry if both situations occurred (e.g. if I stopped that but still got punched). But in others, although it might be surprising, you wouldn t think there was a breach of contract (e.g. if both Mom and Dad took me to the mall). In the LFM, A or B always means A, or B, or perhaps both.

5 Math Propositions and Their Combinations Fall 2003 Page 5 The following statements are true: 7 is an odd number, or 7 is prime. 6 < 9 or 6 > 9. 5 is negative, or -5 is negative. A square is a rectangle or a square is a rhombus. Very often, even in LFM, the two propositions connected by an or will be mutually exclusive, but they don t need to be.

6 Math Propositions and Their Combinations Fall 2003 Page 6 Example: IF THEN Suppose I tell my daughter, If you do that once more you re grounded. Under what conditions can I be assured I am telling the truth? What if 1. She does it once more and I ground her? 2. She doesn t do it once more and I don t ground her? 3. She doesn t do it once more and I ground her? 4. She does it once more and I don t ground her? What we usually mean when we say something like this is that you do that once more and you re grounded will either both be true or both be false.

7 Math Propositions and Their Combinations Fall 2003 Page 7 But let s imagine another situation. I tell my son, If you mow the lawn, I ll take you to Café Rio. Now, what if: 1. He mows the lawn and I take him to Café Rio? 2. He doesn t mow the lawn and I don t take him to Café Rio? 3. He mows the lawn and I don t take him to Café Rio? 4. He doesn t mow the lawn and I take him to Café Rio? It is case #4 that distinguishes between OEE and LFM. In OEE, we would assume that if case 4 happens, my original statement would be a lie. However, from the point of view of my son, case #4 is just fine. He won t accuse me of breach of contract. It may be bad parenting, but it is not a lie. That is how If A, then B is always interpreted in LFM. This takes some getting used to.

8 Math Propositions and Their Combinations Fall 2003 Page 8 Example: AND Luckily, the conjunction and in OEE and the connector and in LFM have pretty much the same meaning. There s not much more to say about it. Example: IF AND ONLY IF There really isn t an OEE equivalent to what if and only if means in LFM. We usually mean I will give you a reward if and only if you clean your room when we say If you clean your room I ll give you a reward. However, there aren t many (any?) everyday expressions that mean if and only if. When we say, A number is prime if and only if it has exactly two factors we mean that for a particular number, being prime and having exactly two factors will either both be true or neither will be true. Another way of saying this is that if a number is prime, it will have exactly two factors, and if a number has exactly two factors, it will be prime. So, A if and only if B says the same thing as if A, then B, and if B, then A. This is, in fact, how we usually approach the proof of an if and only if statement by proving two if-then statements.