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Discrete Mathematical Structures

Dan Yasaki, Sebastian Pauli

Subsection 1.3.1 Predicates

Statements involving variables, such as “\(x\) is greater than \(3\)” are often found in mathematical assertions. Such statements are not propositions as they are neither true nor false when the value of the variable is not specified. The statement has two parts. The first part, the variable \(x\) , is the subject . The second part, the predicate , is the property that the subject can have. In this case, the predicate is “is greater than \(3\)”. We denote such statements as \(P(x)\) , which we read as the propositional function \(P\) evaluated at \(x\text{.}\)
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Definition 1.3.1.1.

A propositional function \(P\) is a function which for each input outputs a proposition.
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Example 1.3.1.2. Propositional function \(x \gt 3\).

Let \(P(x)\) be the statement “\(x \gt 3\)”.
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Then \(P(2)\) is the proposition “\(2 \gt 3\) ,” which is false.
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The proposition \(P(4)\) is the statement “\(4 \gt 3\)”, which is true.
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Propositional functions can have many variables:
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Example 1.3.1.3. Propositional function with three variables.

Let \(P(x,y,z)\) be “\(x\text{,}\) \(y\text{,}\) and \(z\) live in the same dorm.”.
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Then \(P(\text{Alice}, \text{Bob}, \text{Carl})\) is the proposition “Alice, Bob, and Carl live in the same dorm.”
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Definition 1.3.1.4.

The collection of all inputs that we allow for \(P\) is the domain of the propositional function \(P\) .
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Let \(x\) be in the domain of a propositional function \(P\text{.}\) Then \(P(x)\) is a proposition. The property of \(x\) given by the proposition \(P(x)\) is called the predicate of \(P(x)\text{.}\)
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Definition 1.3.1.5.

The integers are the numbers:
\begin{equation*} \dots, -3, -2, -1, 0, 1, 2, 3, \dots \end{equation*}
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Example 1.3.1.6. Predicate.

Let \(P(x)=\) “\(x\lt 5\)” with domain the integers.
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The predicate of \(P\) is “is less than five” or “\(\lt 5\)”.
  1. \(P(4)\) is “\(4\lt 5\)” which is true.
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  2. \(P(10)\) is “\(10\lt 5\)” which is false.
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Example 1.3.1.7. Determine whether \(Q(1)\) and \(Q(2)\) are true.

Let \(Q(x)\) be “\((x+3)^3-25=(5x)^2\)”.
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Determine whether
  1. \(Q(1)\) is true or false and
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  2. \(Q(2)\) is true or false.
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Answer.
\(Q(1)\) is false and \(Q(2)\) is true.
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Solution.
  1. We first determine whether \(Q(1)\) is true or false.
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    For \(x=1\) we have that the left hand side of “\(Q(x)\)” is
    \begin{equation*} (x+3)^3-25=(1+3)^3-25=4^3-25=64-25=39 \end{equation*}
    and the right hand side of “\(Q(x)\)” is
    \begin{equation*} (5x)^2=5^2=25. \end{equation*}
    Because \(39\ne 25\) we conclude that \(Q(1)\) is false.
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  2. We now determine whether \(Q(2)\) is true or false.
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    For \(x=2\) we have that the left hand side of “\(Q(x)\)” is
    \begin{equation*} (x+3)^3-25=(2+3)^3-25=5^3-25=125-25=100 \end{equation*}
    and the right hand side of “\(Q(x)\)” is
    \begin{equation*} (5x)^2=10^2=100. \end{equation*}
    Because the left hand side and the right hand side of \(Q(2)\) are equal we conclude that \(Q(2)\) is true.
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Example 1.3.1.8. Prove \(P(1)\) and \(P(2)\) are true.

Let \(P(n)\) be the proposition “\(1 + 2 + \dots + n = \frac{n(n + 1)}{2}\) .”
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Prove \(P(1)\) and \(P(2)\) are true.
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Answer.
\(P(1)\) is the statement “\(1 = \frac{1(1 + 1)}{2}\text{.}\)” This is true since the right side is
\begin{equation*} \frac{1(1 + 1)}{2} = \frac{2}{2} = 1, \end{equation*}
which is equal to the left side.
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Similarly, \(P(2)\) is the statement “\(1 + 2 = \frac{2(2 + 1)}{2}\text{.}\)” This is true since the left side is
\begin{equation*} 1 + 2 = 3, \end{equation*}
and the right side is
\begin{equation*} \frac{2(2 + 1)}{2} = \frac{6}{2} = 3. \end{equation*}
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