1.2: Propositions and Logic
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Logic is, basically, the study of valid reasoning. When searching the internet, we use Boolean logic – terms like “and” and “or” – to help us find specific web pages that fit in the sets we are interested in. After exploring this form of logic, we will look at logical arguments and how we can determine the validity of a claim.
Boolean Logic
We can often classify items as belonging to sets. If you went the library to search for a book and they asked you to express your search using unions, intersections, and complements of sets, that would feel a little strange. Instead, we typically using words like “and”, “or”, and “not” to connect our keywords together to form a search. These words, which form the basis of Boolean logic, are directly related to our set operations. (Boolean logic was developed by the 19th-century English mathematician George Boole.)
Boolean Logic
Boolean logic combines multiple statements that are either true or false into an expression that is either true or false.
In connection to sets, a search is true if the element is part of the set.
Suppose M is the set of all mystery books, and C is the set of all comedy books. If we search for “mystery”, we are looking for all the books that are an element of the set M; the search is true for books that are in the set.
When we search for “mystery and comedy”, we are looking for a book that is an element of both sets, in the intersection. If we were to search for “mystery or comedy”, we are looking for a book that is a mystery, a comedy, or both, which is the union of the sets. If we searched for “not comedy”, we are looking for any book in the library that is not a comedy, the complement of the set C.
Connection to Set Operations
A and B elements in the intersection A ⋂ B
A or B elements in the union A ⋃ B
not A elements in the complement A c
Notice here that or is not exclusive. This is a difference between the Boolean logic use of the word and common everyday use. When your significant other asks “do you want to go to the park or the movies?” they usually are proposing an exclusive choice – one option or the other, but not both. In Boolean logic, the or is not exclusive – more like being asked at a restaurant “would you like fries or a drink with that?” Answering “both, please” is an acceptable answer.
Example \(\PageIndex{1}\)
Suppose we are searching a library database for Mexican universities. Express a reasonable search using Boolean logic.
Solution
We could start with the search “Mexico and university”, but would be likely to find results for the U.S. state New Mexico. To account for this, we could revise our search to read:
Mexico and university not “New Mexico”
In most internet search engines, it is not necessary to include the word and ; the search engine assumes that if you provide two keywords you are looking for both. In Google’s search, the keyword or has be capitalized as OR, and a negative sign in front of a word is used to indicate not . Quotes around a phrase indicate that the entire phrase should be looked for. The search from the previous example on Google could be written:
Mexico university -“New Mexico”
Example \(\PageIndex{2}\)
Describe the numbers that meet the condition: even and less than 10 and greater than 0
Solution
The numbers that satisfy all three requirements are {2, 4, 6, 8}
Sometimes statements made in English can be ambiguous. For this reason, Boolean logic uses parentheses to show precedent, just like in algebraic order of operations.
Example \(\PageIndex{3}\)
Describe the numbers that meet the condition: odd number and less than 20 and greater than 0 and (multiple of 3 or multiple of 5)
Solution
The first three conditions limit us to the set {1, 3, 5, 7, 9, 11, 13, 15, 17, 19}
The last grouped conditions tell us to find elements of this set that are also either a multiple of 3 or a multiple of 5. This leaves us with the set {3, 5, 9, 15}
Notice that we would have gotten a very different result if we had written (odd number and less than 20 and greater than 0 and multiple of 3) or multiple of 5
The first grouped set of conditions would give {3, 9, 15}. When combined with the last condition, though, this set expands without limits:
{3, 5, 9, 15, 20, 25, 30, 35, 40, 45, …}
Example \(\PageIndex{4}\)
The English phrase “Go to the store and buy me eggs and bagels or cereal” is ambiguous; it is not clear whether the requestors is asking for eggs always along with either bagels or cereal, or whether they’re asking for either the combination of eggs and bagels, or just cereal.
Solution
For this reason, using parentheses clarifies the intent:
Eggs and (bagels or cereal) means Option 1: Eggs and bagels, Option 2: Eggs and cereal
(Eggs and bagels) or cereal means Option 1: Eggs and bagels, Option 2: Cereal
Be aware that when a string of conditions is written without grouping symbols, it is often interpreted from the left to right, resulting in the latter interpretation.
Conditional Statements
Beyond searching, Boolean logic is commonly used in spreadsheet applications like Excel to do conditional calculations. A statement is something that is either true or false. A statement like 3 < 5 is true; a statement like “a rat is a fish” is false. A statement like “x < 5” is true for some values of x and false for others. When an action or outcome depends on the value of a statement, it forms a conditional.
Statements and Conditionals
A statement is either true or false. A conditional is a compound statement of the form “if p then q” or “if p then q, else s”.
Example \(\PageIndex{5}\)
In common language, an example of a conditional statement would be “If it is raining, then we’ll go to the mall. Otherwise we’ll go for a hike.”
Solution
The statement “If it is raining” is the condition – this may be true or false for any given day. If the condition is true, then we will follow the first course of action and go to the mall. If the condition is false, then we will use the alternative and go for a hike.
Example \(\PageIndex{6}\)
As mentioned earlier, conditional statements are commonly used in spreadsheet applications like Excel or Google Sheets.
In Excel, you can enter an expression like =IF(A1<2000, A1+1, A1*2)
Solution
Notice that after the IF, there are three parts. The first part is the condition, and the second two are calculations. Excel will look at the value in cell A1 and compare it to 2000. If that condition is true, then the first calculation is used, and 1 is added to the value of A1 and the result is stored. If the condition is false, then the second calculation is used, and A1 is multiplied by 2 and the result is stored.
In other words, this statement is equivalent to saying “If the value of A1 is less than 2000, then add 1 to the value in A1. Otherwise, multiply A1 by 2”.
Example \(\PageIndex{7}\)
The expression =IF(A1>5, 2*A1, 3*A1) is used. Find the result if A1 is 3, and the result if A1 is 8.
Solution
This is equivalent to saying
If A1 > 5, then calculate 2*A1. Otherwise, calculate 3*A1
If A1 is 3, then the condition is false, since 3 > 5 is not true, so we do the alternate action, and multiply by 3, giving 3*3 = 9
If A1 is 8, then the condition is true, since 8 > 5, so we multiply the value by 2, giving 2*8=16
Example \(\PageIndex{8}\)
An accountant needs to withhold 15% of income for taxes if the income is below $30,000, and 20% of income if the income is $30,000 or more. Write an expression that would calculate the amount to withhold.
Solution
Our conditional needs to compare the value to 30,000. If the income is less than 30,000, we need to calculate 15% of the income: 0.15*income. If the income is more than 30,000, we need to calculate 20% of the income: 0.20*income.
In words we could write “If income < 30,000, then multiply by 0.15, otherwise multiply by 0.20”. In Excel, we would write:
=IF(A1<30000, 0.15*A1, 0.20*A1)
As we did earlier, we can create more complex conditions by using the operators and, or , and not to join simpler conditions together.
Example \(\PageIndex{9}\)
A parent might say to their child “if you clean your room and take out the garbage, then you can have ice cream.”
Here, there are two simpler conditions:
1) The child cleaning her room
2) The child taking out the garbage
Solution
Since these conditions were joined with and , the combined conditional will be true only if both simpler conditions are true; if either chore is not completed, then the parent’s condition is not met.
Notice that if the parent had said “if you clean your room or take out the garbage, then you can have ice cream”, then the child would need to complete only one chore to meet the condition.
Suppose you wanted to have something happen when a certain value is between 100 and 300. To create the condition “A1 < 300 and A1 > 100” in Excel, you would need to enter “AND(A1<300, A1>100)”. Likewise, for the condition “A1=4 or A1=6” you would enter “OR(A1=4, A1=6)”
Example \(\PageIndex{10}\)
In a spreadsheet, cell A1 contains annual income, and A2 contains number of dependents.
A certain tax credit applies if someone with no dependents earns less than $10,000, or if someone with dependents earns less than $20,000. Write a rule that describes this.
Solution
There are two ways the rule is met:
income is less than 10,000 and dependents is 0, or
income is less than 20,000 and dependents is not 0.
Informally, we could write these as
(A1 < 10000 and A2 = 0) or (A1 < 20000 and A2 > 0)
In Excel’s format, we’d write
IF(OR(AND(A1<10000, A2=0), AND(A1<20000, A2>0)), “you qualify”, “you don’t qualify”)
Quantified Statements
Words that describe an entire set, such as “all”, “every”, or “none”, are called universal quantifiers because that set could be considered a universal set. In contrast, words or phrases such as “some”, “one”, or “at least one” are called existential quantifiers because they describe the existence of at least one element in a set.
Quantifiers
A universal quantifier states that an entire set of things share a characteristic.
An existential quantifier states that a set contains at least one element.
Something interesting happens when we negate – or state the opposite of – a quantified statement.
Example \(\PageIndex{11}\)
Suppose your friend says “Everybody cheats on their taxes.” What is the minimum amount of evidence you would need to prove your friend wrong?
Solution
To show that it is not true that everybody cheats on their taxes, all you need is one person who does not cheat on their taxes. It would be perfectly fine to produce more people who do not cheat, but one counterexample is all you need.
It is important to note that you do not need to show that absolutely nobody cheats on their taxes.
Example \(\PageIndex{12}\)
Suppose your friend says “One of these six cartons of milk is leaking.” What is the minimum amount of evidence you would need to prove your friend wrong?
Solution
In this case, you would need to check all six cartons and show that none of them is leaking. You cannot disprove your friend’s statement by checking only one of the cartons.
When we negate a statement with a universal quantifier, we get a statement with an existential quantifier, and vice-versa.
Negating a quantified statement
The negation of “all A are B” is “at least one A is not B”.
The negation of “no A are B” is “at least one A is B”.
The negation of “at least one A is B” is “no A are B”.
The negation of “at least one A is not B” is “all A are B”.
Example \(\PageIndex{13}\)
“Somebody brought a flashlight.” Write the negation of this statement.
Solution
The negation is “Nobody brought a flashlight.”
Example \(\PageIndex{14}\)
“There are no prime numbers that are even.” Write the negation of this statement.
Solution
The negation is “At least one prime number is even.”
Try it Now 1
- Write the negation of “All Icelandic children learn English in school.”
Contributors and Attributions
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Saburo Matsumoto
CC-BY-4.0