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Chapter 5: Problem 25

Suppose that \(P(n)\) is a propositional function. Determine for which positive integers \(n\) the statement \(P(n)\) must be true, and justify your answer, if a) \(P(1)\) is true; for all positive integers \(n,\) if \(P(n)\) is true, then \(P(n+2)\) is true. b) \(P(1)\) and \(P(2)\) are true; for all positive integers \(n,\) if \(P(n)\) and \(P(n+1)\) are true, then \(P(n+2)\) is true. c) \(P(1)\) is true; for all positive integers \(n,\) if \(P(n)\) is true, then \(P(2 n)\) is true. d) \(P(1)\) is true; for all positive integers \(n,\) if \(P(n)\) is true, then \(P(n+1)\) is true.

Short Answer

Expert verified
a) True for all odd positive integers. b) True for all positive integers. c) True for powers of 2. d) True for all positive integers.

Step by step solution

01

Part a) Check Base Case and Induction Step

The base case is given as \(P(1) = \text{true}\). The induction step states that for all positive integers \(n\), if \(P(n)\) is true, then \(P(n+2)\) is true. Therefore, \(P(1) \rightarrow P(3) \rightarrow P(5) \rightarrow \text{and so on}\). This implies \(P(n)\) is true for all odd positive integers.
02

Part b) Check Base Cases and Induction Step

The base cases given are \(P(1) = \text{true}\) and \(P(2) = \text{true}\). The induction step states that for all positive integers \(n\), if both \(P(n)\) and \(P(n+1)\) are true, then \(P(n+2)\) is true. Using this step \(P(1) \text{ and } P(2) \rightarrow P(3), P(2) \text{ and } P(3) \rightarrow P(4), \text{ and so on}\). This implies \(P(n)\) is true for all positive integers.
03

Part c) Check Base Case and Induction Step

The base case is given as \(P(1) = \text{true}\). The induction step states that for all positive integers \(n\), if \(P(n)\) is true, then \(P(2n)\) is true. Therefore, \(P(1) \rightarrow P(2) \rightarrow P(4) \rightarrow P(8), \text{ and so on}\). Thus, \(P(n)\) is true for all powers of 2.
04

Part d) Check Base Case and Induction Step

The base case is given as \(P(1) = \text{true}\). The induction step states that for all positive integers \(n\), if \(P(n)\) is true, then \(P(n+1)\) is true. Using this step \(P(1) \rightarrow P(2) \rightarrow P(3) \rightarrow \text{and so on}\). This implies \(P(n)\) is true for all positive integers.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Propositional Function
In mathematics, a propositional function is a statement containing variables that becomes a proposition when the variables are assigned specific values. For instance, think of a statement like 'x is greater than 2.' This is not just a simple proposition because x is a variable and the truth of the statement depends on the value of x.

When studying propositional functions in discrete mathematics, we often look at whether they are true under certain conditions. In induction problems, we typically use these functions in a sequence, testing different values to establish a pattern or a rule.
Proof Techniques
Mathematical proofs are logical arguments that establish the truth of a given statement. Several key proof techniques are commonly used, including mathematical induction.

Mathematical induction is particularly useful for proving statements about sequences or series. It involves two main steps:
  • Base Case: Prove the statement is true for an initial value, usually the smallest value in the domain.
  • Inductive Step: Assume the statement is true for a specific case, say 'n=k.' Then show it holds true for the next case, 'n=k+1.' This helps us infer that the statement is true for all subsequent values.
For example, in Part d) of the exercise, we start with \(P(1)=\text{true}\) and then show that if \(P(n)\) is true, \(P(n+1)\) must also be true. This stepwise validation proves that the original statement holds for all positive integers.

Other proof techniques include direct proof, proof by contradiction, and combinatorial proofs, which each have their unique applications depending on what you are trying to prove.
Discrete Mathematics
Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. This includes objects like integers, graphs, and statements in logic.

Some of the key areas within discrete mathematics include:
  • Logic and Propositions: Understanding the logical statements and their truth values.
  • Set Theory: Study of collections of objects.
  • Combinatorics: Counting methods and the study of different arrangements of objects.
  • Graph Theory: Study of graphs and networks.
  • Algorithms: Step-by-step procedure for calculations and problem-solving.
In solving the given exercise, we use propositional functions and induction, both of which are pivotal areas of discrete mathematics. This branch of mathematics lays a strong foundation for computer science and other fields where algorithms and discrete structures are crucial.

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Most popular questions from this chapter

Prove that the program segment $$ \begin{array}{l}{y :=1} \\ {z :=x+y}\end{array} $$ is correct with respect to the initial assertion \(x=0\) and the final assertion \(z=1\)

Suppose that \(P\) is a simple polygon with vertices \(v_{1}, v_{2}, \ldots, v_{n}\) listed so that consecutive vertices are con- nected by an edge, and \(v_{1}\) and \(v_{n}\) are connected by an edge. A vertex \(v_{i}\) is called an ear if the line segment connecting the two vertices adjacent to \(v_{i}\) is an interior diagonal of the simple polygon. Two ears \(v_{i}\) and \(v_{j}\) are called nonoverlapping if the interiors of the triangles with vertices \(v_{i}\) and its two adjacent vertices and \(v_{j}\) and its two adjacent vertices do not intersect. Prove that every simple polygon with at least four vertices has at least two nonoverlapping ears.

Show that a three-dimensional \(2^{n} \times 2^{n} \times 2^{n}\) checkerboard with one \(1 \times 1 \times 1\) cube missing can be completely covered by \(2 \times 2 \times 2\) cubes with one \(1 \times 1 \times 1\) cube removed.

Let \(a_{1}, a_{2}, \ldots, a_{n}\) be positive real numbers. The arithmetic mean of these numbers is defined by $$ A=\left(a_{1}+a_{2}+\cdots+a_{n}\right) / n $$ and the geometric mean of these numbers is defined by $$ G=\left(a_{1} a_{2} \cdots a_{n}\right)^{1 / n} . $$ Use mathematical induction to prove that \(A \geq G\) .

Use mathematical induction in Exercises \(38-46\) to prove results about sets. Prove that a set with \(n\) elements has \(n(n-1) / 2\) subsets containing exactly two elements whenever \(n\) is an integer greater than or equal to \(2 .\)
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