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Lesson 145: Mathematical Logic Using Symbols
In this lesson we'll continue the study of mathematical logic. First we learned the material using real sentences, and in this lesson we'll use symbols. Note that not all schools teach this topic. Find out if your school does, and if you will be tested on it.
Before studying this lesson, be sure that you have studied the previous lesson, even if it does not fully make sense, or is hard to memorize.
In the last lesson, we worked with real English sentences. In this lesson, we will use variables to represent sentences. The most common choice of letters for this purpose are p and q. Recall that the negation is the opposite of a statement. Given a statement p, the negation can be represented as ~p. The tilde (~) is read as "not."
We can make a truth table showing both possible true-false values of p, and what the corresponding value of ~p would be:
| p | ~p |
| T | F |
| F | T |
Recall that a conjunction is a compound statement that is formed by joining two statements with the word "and." We could represent this as p ^ q. The symbol ^ is read as "and."
We can make a truth table showing all four possible true-false combinations of p and q, and what the corresponding value of p ^ q would be. Notice that a conjunction is only true when p and q are both true.
| p | q | p^q |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
Recall that a disjunction is a compound statement that is formed by joining two statements with the word "or." We could represent this as p v q. The symbol v is read as "or". It has nothing to do with the letter "v", it's just an upside-down "and" symbol.
We can make a truth table showing all four possible true-false combinations of p and q, and what the corresponding value of p v q would be. Notice that a disjunction is only false when p and q are both false.
| p | q | p v q |
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
Recall that a conditional is a compound statement that is formed by joining two statements with the words "if...then." We could represent this as p → q, read as "if p, then q."
We can make a truth table showing all four possible true-false combinations of p and q, and what the corresponding value of p → q would be. Notice that a conditional is only false when p is true, and q is false.
| p | q | p → q |
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
The converse of p → q would be
written symbolically as q → p
The inverse of p → q would be
written symbolically as ~p → ~q
The contrapositive of p → q would
be written symbolically as ~q → ~p
We can make a very big truth table showing all four possible true-false combinations of p and q, and what the corresponding value of each of these variations would be. Study the table carefully, and make sure that you understand how all of the values were obtained.
| p | q | ~p | ~q | p → q | q → p | ~p → ~q | ~q → ~p |
| T | T | F | F | T | T | T | T |
| T | F | F | T | F | T | T | F |
| F | T | T | F | T | F | F | T |
| F | F | T | T | T | T | T | T |
Look at the columns for p → q and ~q → ~p. They are identical. We can say that a conditional and it's contrapositive are logically equivalent.
Recall that a biconditional is a statement that is formed by joining two statements with the words "if and only if". We can represent this as p ↔ q.
We can make a truth table showing all four true-false combinations of p and q, and what the corresponding value of p ↔ q would be. Notice that the biconditional is true when p and q are both false, or both true, and it is false then the values of p and q are opposite.
| p | q | p ↔ q |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
Recall that a tautology is a compound statement that is always true. Symbolically a statement is a tautology if its value is true for all four rows of its truth table. Stated another way, the statement is true no matter what truth values of p and q we use. Stated yet another way looking down the column of a tautology, we will see T T T T.
This is all tricky, and requires lots of memorization and practice. There is no other way to learn this topic.
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