Back To CourseMath 102: College Mathematics
15 chapters | 122 lessons | 13 flashcard sets
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Amy has a master's degree in secondary education and has taught math at a public charter high school.
Maria has a blue dog. Kevin has a purple cat. Joann has a black rat. These are examples of propositions. Propositions, in logic, are statements that can be labeled as either true or false. All of the examples you just heard and saw are complete statements that you can say are either true or false.
In logic, we sometimes change our original statement to its negative form. We do this by adding a NOT in the statement. So, the negative of 'Maria has a blue dog' is 'Maria does not have a blue dog.' We've added a few words just to make it grammatically correct, but as you can see, we have added a NOT in the statement. We went from stating that something is happening to something that is not happening.
If our original proposition is in the negative form, then the negative form of that statement will be a positive. For example, if our original statement is 'We are not in the year 1990,' then the negative of that statement becomes 'We are in the year 1990.'
Think of the negative as adding a NOT if there is no NOT and deleting the NOT if there is a NOT. You are essentially turning a positive into a negative or a negative into a positive depending on what kind of statement you begin with.
Negating a proposition changes its truth value, whether the statement is true or false. For example, if the statement 'She loves to chase squirrels' is true, then the negative of the statement, 'She does not love to chase squirrels,' is false.
We can create a simple table to show the truth value of a statement and its negation. We will call our statement p and the negation NOT p. We write these in the top row of our truth value table. In the next row, we put T under the p column. Now, if the statement p is true, then its negation NOT p must be false, so we put F in the same row under the NOT p column. In the next row, we put F under the p column and T under the NOT p column since if our original statement is false, then the negation must be true. We can use this truth value table for any logic proposition we come across.
We can look at any proposition and compare it to this truth value table. If our original proposition is false, then its negation is true. If our original proposition is true, then its negation is false. For example, if we know the proposition '2 + 2 = 5' is false, then by looking at the third row in the chart, we can see that the negation '2 + 2 does not = 5' is true.
We can take our truth value table one step further by adding a second proposition into the mix. By adding a second proposition and including all the possible scenarios of the two propositions together, we create a truth table, a table showing the truth value for logic combinations. We will call our first proposition p and our second proposition q. With just these two propositions, we have four possible scenarios.
We can have both statements true; we can have the first statement true and the second false; we can have the first statement false and the second true; and we can have both statements false. For example, if our first proposition, p, is 'Ed is a horse,' and our second proposition, q, is 'Spot is a dog,' then we can have four possible scenarios by combining these two statements. We can have both statements being true, we can have the first statement being true and the second statement false, we can have the first statement being false and the second true, or we can have both statements being false.
The truth table sets all these scenarios up so you can quickly look up your situation to find its truth value. This is just the beginning of our truth table where we set up our scenarios. Keep watching and you will see how to include the truth values for the logic combinations.
There are four logical combinations we can make with these two statements.
1.p AND q. When you have an AND connecting the two simple statements, it means that both statements must be happening at the same time. For example, if our first proposition is 'The room is blue' and our second proposition is 'The lamp is blue,' then p AND q means that both the room and the lamp are blue. In order for this type of 'and statement' to be true, both statements must be true to begin with. We can show this by adding a column to our truth table for p AND q and labeling the row where both p AND q are true with a T and the rest with an F.
|p||q||p AND q|
2.p OR q. When you combine the two propositions with an OR, it means that either or both is happening. If our first proposition is 'The cat is chasing the mouse' and our second proposition is 'The dog is chasing the cat,' combining the two with an OR means that we can see the cat chasing the mouse or we can see the dog chasing the cat or we can see both the dog chasing the cat and the cat chasing the mouse. For this case, if just one of the statements is true, the OR statement will be true. Let's add this information to our truth table under the column p OR q. The only scenario when this is false is when both statements are false to begin with.
|p||q||p AND q||p OR q|
3.If p, then q. In this case, the second proposition will happen if the first proposition happens. For example, if our first proposition is 'Jimmy loses a tooth' and our second proposition is 'Jimmy finds a dollar,' combining the two in this way means that if Jimmy loses a tooth is true, then Jimmy finds a dollar is also true. If Jimmy doesn't find a dollar, then this combination is false. If Jimmy doesn't lose a tooth, whether he finds a dollar or not is irrelevant and either case will be true for this combination. Think of this as a kind of promise. The only way to break a promise and make this combination false is if the first proposition happens and you don't fulfill the second proposition. All other cases will be true.
|p||q||p AND q||p OR q||If p, then q|
4.p if and only if q. This last combination means that either proposition happens only if the other proposition happens. So, if my first proposition is 'We will go to the amusement park' and my second proposition is 'We will go to the zoo,' this combination tells me that either we go to the amusement park and the zoo or we go to neither. We can't have one without the other. To fill our truth table for this combination, we mark a T for when both statements are either true or false.
|p||q||p AND q||p OR q||If p, then q||p if and only if q|
We can use this truth table to find the truth value for the AND, OR, if-then, and if and only if logic combinations of two propositions by looking up our scenario first and then finding our logic combination.
We can create our own truth table for combinations of three propositions or more by adding more rows and columns to account for more propositions and scenarios. For two propositions, we only have four scenarios. For three propositions, our scenarios jump to eight since we are adding another proposition that can be either true or false.
From this point on, we can build on to our truth table with the various combinations that we need.
In review, we have learned that propositions are statements that can be labeled as either true or false. We have also learned that the truth value of a statement is whether it is true or false and a truth table is a table showing all the truth values for logic combinations. The four logic combinations that we have discussed are AND, OR, if-then, and if and only if. AND means that both statements must be true for the combination to be true. OR means that either statement must be true for the combination to be true. If-then means that the second statement must happen when the first statement happens. If and only if means that both statements must be either true or false for the combination to be true.
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Back To CourseMath 102: College Mathematics
15 chapters | 122 lessons | 13 flashcard sets