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Compound Statements

Translate compound statements into symbolic form.

Learning Objectives

After completing this section, you should be able to:

  1. Translate compound statements into symbolic form.
  2. Translate compound statements in symbolic form with parentheses into words.
  3. Apply the dominance of connectives.

Common Logical Connectives

Understanding the following logical connectives, along with their properties, symbols, and names, will be key to applying the topics presented in this chapter. The chapter will discuss each connective introduced here in more detail.

The joining of two logical statements with the word "and" or "but" forms a compound statement called a conjunction. In logic, for a conjunction to be true, all the independent logical statements that make it up must be true. The symbol for a conjunction is \(∧\). Consider the compound statement, “Derek Jeter played professional baseball for the New York Yankees, and he was a shortstop.” If \(p\) represents the statement, “Derrick Jeter played professional baseball for the New York Yankees,” and if \(q\) represents the statement, “Derrick Jeter was a short stop,” then the conjunction will be written symbolically as \(p∧q.\)

The joining of two logical statements with the word “or” forms a compound statement called a disjunction. Unless otherwise specified, a disjunction is an inclusive or statement, which means the compound statement formed by joining two independent clauses with the word or will be true if a least one of the clauses is true. Consider the compound statement, "The office manager ordered cake for for an employee’s birthday or they ordered ice cream.” This is a disjunction because it combines the independent clause, “The office manager ordered cake for an employee’s birthday,” with the independent clause, “The office manager ordered ice cream,” using the connective, or. This disjunction is true if the office manager ordered only cake, only ice cream, or they ordered both cake and ice cream. Inclusive or means you can have one, or the other, or both!

Joining two logical statements with the word implies, or using the phrase “if first statement, then second statement,” is called a conditional or implication. The clause associated with the "if" statement is also called the hypothesis or antecedent, while the clause following the "then" statement or the word implies is called the conclusion or consequent. The conditional statement is like a one-way contract or promise. The only time the conditional statement is false, is if the hypothesis is true and the conclusion is false. Consider the following conditional statement, “If Pedro does his homework, then he can play video games.” The hypothesis/antecedent is the statement following the word if, which is “Pedro does/did his homework.” The conclusion/consequent is the statement following the word then, which is “Pedro can play his video games.”

Condensed — the full section is in OpenStax Contemporary Mathematics.

Translating Compound Statements to Symbolic Form

To translate a compound statement into symbolic form, we take the following steps.

  1. Identify and label all independent affirmative logical statements with a lower case letter, such as \(p\), \(q\), or \(r\).
  2. Identify and label any negative logical statements with a lowercase letter preceded by the negation symbol, such as \(\sim p\), \(\sim q\), or \(\sim r\).
  3. Replace the connective words with the symbols that represent them, such as \(∧,∨,\to ,\text{or}↔.\)

Consider the previous example of your friend trying to get their driver’s license. Your friend passed the written test, but they did not pass the road test. Let \(p\) represent the statement, “My friend passed the written test.” And, let \(\sim q\) represent the statement, “My friend did not pass the road test.” Because the connective but is logically equivalent to the word and, the symbol for but is the same as the symbol for and; replace but with the symbol \(∧.\) The compound statement is symbolically written as: \(p∧\sim q\). My friend passed the written test, but they did not pass the road test.

Condensed — the full section is in OpenStax Contemporary Mathematics.

Translating Compound Statements in Symbolic Form with Parentheses into Words

When parentheses are written in a logical argument, they group a compound statement together just like when calculating numerical or algebraic expressions. Any statement in parentheses should be treated as a single component of the expression. If multiple parentheses are present, work with the inner most parentheses first.

Consider your friend’s struggles to get their license to drive. Let \(p\) represent the statement, “My friend passed the written test,” let \(q\) represent the statement, “My friend passed the road test,” and let \(r\) represent the statement, “My friend received a driver’s license.” The statement \((p∧q)\to r\) can be translated into words as follows: the statement \(p∧q\) is grouped together to form the hypothesis of the conditional statement and \(r\) is the conclusion. The conditional statement has the form “if \(p∧q,\) then \(r.\)” Therefore, the written form of this statement is: “If my friend passed the written test and they passed the road test, then my friend received a driver’s license.”

Sometimes a compound statement within parentheses may need to be negated as a group. To accomplish this, add the phrase, “it is not the case that” before the translation of the phrase in parentheses. For example, using \(p\), \(q\), and \(r\) of your friend obtaining a license, let’s translate the statement \(\sim (p∧q)\to \sim r\) into words.

In this case, the hypothesis of the conditional statement is \(\sim (p∧q)\) and the conclusion is \(\sim r.\) To negate the hypothesis, add the phrase “it is not the case” before translating what is in parentheses. The translation of the hypothesis is the sentence, “It is not the case that my friend passed the written test and they passed the road test,” and the translation of the conclusion is, “My friend did not receive a driver’s license.” So, a translation of the complete conditional statement, \(\sim (p∧q)\to \sim r\) is: “If it is not the case that my friend passed the written test and the road test, then my friend did not receive a driver’s license.”

Condensed — the full section is in OpenStax Contemporary Mathematics.

The Dominance of Connectives

The order of operations for working with algebraic and arithmetic expressions provides a set of rules that allow consistent results. For example, if you were presented with the problem \(1+3\times 2\), and you were not familiar with the order of operation, you might assume that you calculate the problem from left to right. If you did so, you would add 1 and 3 to get 4, and then multiply this answer by 2 to get 8, resulting in an incorrect answer. Try inputting this expression into a scientific calculator. If you do, the calculator should return a value of 7, not 8.

Scientific Calculator

The order of operations for algebraic and arithmetic operations states that all multiplication must be applied prior to any addition. Parentheses are used to indicate which operation—addition or multiplication—should be done first. Adding parentheses can change and/or clarify the order. The parentheses in the expression \(1+(3\times 2)\) indicate that 3 should be multiplied by 2 to get 6, and then 1 should be added to 6 to get 7: \(1+(3\times 2)=7.\)

As with algebraic expressions, there is a set of rules that must be applied to compound logical statements in order to evaluate them with consistent results. This set of rules is called the dominance of connectives. When evaluating compound logical statements, connectives are evaluated from least dominant to most dominant as follows:

  1. Parentheses are the least dominant connective. So, any expression inside parentheses must be evaluated first. Add as many parentheses as needed to any statement to specify the order to evaluate each connective.
  2. Next, we evaluate negations.
  3. Then, we evaluate conjunctions and disjunctions from left to right, because they have equal dominance.
  4. After evaluating all conjunctions and disjunctions, we evaluate conditionals.
  5. Lastly, we evaluate the most dominant connective, the biconditional. If a statement includes multiple connectives of equal dominance, then we will evaluate them from left to right.

See for a visual breakdown of the dominance of connectives.

Let’s revisit your friend’s struggles to get their driver’s license. Let \(p\) represent the statement, “My friend passed the written test,” let \(q\) represent the statement, “My friend passed the road test,” and let \(r\) represent the statement, “My friend received a driver’s license.” Let's use the dominance of connectives to determine how the compound statement \(p∧\sim q\to r\) should be evaluated.

Step 1: There are no parentheses, which is least dominant of all connectives, so we can skip over that.

Condensed — the full section is in OpenStax Contemporary Mathematics.

Key Concepts

  • Logical connectives are used to form compound logical statements by using words such as and, or, and if …, then.
  • A conjunction is a compound logical statement formed by combining two statements with the words “and” or “but.” If the two independent clauses are represented by \(p\) and \(q\), respectively, then the conjunction is written symbolically as \(p∧q\). For the conjunction to be true, both \(p\) and \(q\) must be true.
  • A disjunction joins two logical statements with the or connective. In, logic or is inclusive. For an or statement to be true at least one statement must be true, but both may also be true.
  • A conditional statement has the form if \(p\), then \(q\), where \(p\) and \(q\) are logical statements. The only time the conditional statement is false is when \(p\) is true, and \(q\) is false.
  • The biconditional statement is formed using the connective \(if and only if\) for the biconditional statement to be true, the true values of \(p\) and \(q\), must match. If \(p\) is true then \(q\) must be true, if \(p\) is false, then \(q\) must be false.
  • Translate compound statements between words and symbolic form.
    ConnectiveSymbolName
    and
    but    		\(∧\)conjunction
    or\(∨\)disjunction, inclusive or
    not~negation
    if \(\ldots\), then implies\(\to\)conditional, implication
    if and only if\(↔\)biconditional
  • The dominance of connectives explains the order in which compound logical statements containing multiple connectives should be interpreted.
  • The dominance of connectives should be applied in the following order
    • Parentheses
    • Negations
    • Disjunctions/Conjunctions, left to right
    • Conditionals
    • Biconditionals

Practice (4)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

  1. For each of the following connectives, write its name and associated symbol.

    1. or
    2. implies
    3. but
    Afslør svaret
    1. A compound statement formed with the connective word or is called a disjunction, and it is represented by the \(∨\) symbol.
    2. A compound statement formed with the connective word implies or phrase “if …, then” is called a conditional statement or implication and is represented by the \(\to\) symbol.
    3. A compound statement formed with the connective words but or and is called a conjunction, and it is represented by the \(∧\) symbol.

  2. Let \(p\) represent the statement, “It is a warm sunny day,” and let \(q\) represent the statement, “the family will go to the beach.” Write the symbolic form of each of the following compound statements.

    1. If it is a warm sunny day, then the family will go to the beach.
    2. The family will go to the beach, and it is a warm sunny day.
    3. The family will not go to the beach if and only if it is not a warm sunny day.
    4. The family not go to the beach, or it is a warm sunny day.
    Afslør svaret
    1. Replace “it is a warm sunny day” with \(p\). Replace “the family will go to the beach.” with \(q\). Next. Next, because the connective is if …, then place the conditional symbol, \(\to\), between \(p\) and \(q\). The compound statement is written symbolically as: \(p\to q.\)
    2. Replace “The family will go to the beach” with \(q\). Replace “it is a warm sunny day.” with \(p\). Next, because the connective is and, place the \(∧\) symbol between \(q\) and \(p\). The compound statement is written symbolically as: \(q∧p.\)
    3. Replace “The family will not go to the beach. with \(\sim q\). Replace “it is not a warm sunny day” with \(\sim p\). Next, because the connective is or, if and only if, place the biconditional symbol, \(↔\) between \(\sim q\) and \(\sim p\). The compound statement is written symbolically as: \(\sim q↔\sim p.\)
    4. Replace “The family will not go to the beach” with \(\sim q\). Replace “it is a warm sunny day” with \(p\). Next, because the connective is or, place the \(∨\) symbol between \(\sim q\) and \(p\). The compound statement is written symbolically as: \(\sim q∨p.\)

  3. Let \(p\) represent the statement, “My child finished their homework,” let \(q\) represent the statement, “My child cleaned her room,” let \(r\) represent the statement, “My child played video games,” and let \(s\) represent the statement, “My child streamed a movie.” Translate each of the following symbolic statements into words.

    1. \(\sim (p∧q)\)
    2. \((p∧q)\to (r∨s)\)
    3. \(\sim (r∨s)↔\sim (p∧q)\)
    Afslør svaret
    1. Replace ~ with “It is not the case,” and \(∧\) with “and.” One possible translation is: “It is not the case that my child finished their homework and cleaned their room.”
    2. The hypothesis of the conditional statement is, “My child finished their homework and cleaned their room.” The conclusion of the conditional statement is, “My child played video games or streamed a movie.” One possible translation of the entire statement is: “If my child finished their homework and cleaned their room, then they played video games or streamed a movie.”
    3. The hypothesis of the biconditional statement is \(\sim (r∨s)\) and is written in words as: “It is not the case that my child played video games or streamed a movie.” The conclusion of the biconditional statement is \(\sim (p∧q)\), which translates to: “It is not the case that my child finished their homework and cleaned their room.” Because the biconditional, \(↔\) translates to if and only if, one possible translation of the statement is: “It is not the case that my child played video games or streamed a movie if and only if it is not the case that my child finished their homework and cleaned their room.”

  4. For each of the following compound logical statements, add parentheses to indicate the order to evaluate the statement. Recall that parentheses are evaluated innermost first.

    1. \(p∧\sim q∨r\)
    2. \(q\to \sim p∧r\)
    3. \(\sim (p∨q)↔\sim p∧\sim q\)
    Afslør svaret
    1. Because negation is the least dominant connective, we evaluate it first: \(p∧(\sim q)∨r.\) Because conjunction and disjunction have the same dominance, we evaluate them left to right. So, we evaluate the conjunction next, as indicated by the additional set of parentheses: \((p∧(\sim q))∨r.\) The only remaining connective is the disjunction, so it is evaluated last, as indicated by the third set of parentheses. The complete solution is: \(((p∧(\sim q))∨r).\)
    2. Negation has the lowest dominance, so it is evaluated first: \(q\to (\sim p)∧r.\) The remaining connectives are the conditional and the conjunction. Because conjunction has a lower precedence than the conditional, it is evaluated next, as indicated by the second set of parentheses: \(q\to ((\sim p)∧r).\) The last step is to evaluate the conditional, as indicated by the third set of parentheses: \((q\to ((\sim p)∧r)).\)
    3. This statement is known as De Morgan’s Law for the negation of a disjunction. It is always true. Section 2.6 of this chapter will explore De Morgan’s Laws in more detail.
      • First, we evaluate the negations on the right side of the biconditional prior to the conjunction.
      • Then, we evaluate the disjunction on the left side of the biconditional, followed by the negation of the disjunction on the left side.
      • Lastly, after completely evaluating each side of the biconditional, we evaluate the biconditional. It does not matter which side you begin with.
      The final solution is: \((\sim (p∨q))↔((\sim p)∧(\sim q)).\)

Symbols used here

n!
factorial
n × (n−1) × … × 1; the number of orderings of n things. 0! = 1.
\binom{n}{k}
binomial coefficient, "n choose k"
Number of k-element subsets of n things: n!/(k!(n−k)!).
\sum_{k=1}^{n} a_k
summation
Add a_k for k = 1 up to n.
x \in A,\ A \subseteq B
element of, subset
x belongs to A; every element of A is in B.
A \cup B,\ A \cap B,\ A \setminus B
union, intersection, difference
In either; in both; in A but not B.
\emptyset,\ |A|
empty set, cardinality
The set with no elements; the number of elements of A.
\forall,\ \exists
for all, there exists
Quantifiers: every x; at least one x.
\neg,\ \wedge,\ \vee,\ \Rightarrow,\ \Leftrightarrow
not, and, or, implies, iff
Logical connectives.
\blacksquare\ \text{or}\ \square
end of proof (halmos)
Marks the point where the statement has been established.
a \equiv b \pmod n
congruent modulo n
n divides a − b; a and b have the same remainder.
O(n^2),\ \Theta,\ \Omega
big-O notation
Grows no faster than n² (up to a constant), for large n.
a \bmod n
remainder
What is left after dividing a by n.

How to: Compound Statements

  1. Translate compound statements into symbolic form.
  2. Translate compound statements in symbolic form with parentheses into words.
  3. Apply the dominance of connectives.
  4. or
  5. implies
  6. but
  7. A compound statement formed with the connective word
  8. A compound statement formed with the connective word

Questions people ask

What makes mathematics "discrete"?

It deals with separate, countable objects — integers, graphs, statements — rather than continuous quantities. No limits, no infinitesimals; instead induction, counting and logic.

How does a proof by induction work?

Show the statement for the first case, then show that whenever it holds for n it holds for n + 1. Like dominoes: the first falls, and each knocks over the next.

Prøv din egen

Parts of this page are adapted from OpenStax Contemporary Mathematics (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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