Problem 57
Question
Construct a derivation tree for each expression. $$5+(4 \uparrow 3)$$
Step-by-Step Solution
Verified Answer
Construct the given derivation tree as follows:
1. Start with the final evaluated result as the root node: \(69\)
2. Add two child nodes for the addition operation: \(5\) and \(64\)
3. The node for \(64\) becomes the root of another subtree representing the up-arrow operation.
4. Add two more child nodes for this up-arrow notation subtree: \(4\) and \(3\)
The derivation tree looks like:
69
/ \
5 64
/ \
4 3
1Step 1: Identify the operation precedence
In the given expression, there are two operations:
1. Addition
2. Up-arrow notation for exponentiation
We need to follow the order of precedence of these operations, which is as follows:
1. Up-arrow notation
2. Addition
2Step 2: Break down the up-arrow notation
In the given expression, the up-arrow notation is given by \(4 \uparrow 3\). This notation refers to exponentiation, i.e., raising 4 to the power of 3.
So, \(4 \uparrow 3 = 4^3 = 4 \times 4 \times 4 = 64\)
3Step 3: Break down the addition
Now, replace the up-arrow notation with its evaluated result in the original expression.
The expression now becomes: \(5 + 64\)
Add the two numbers: \(5 + 64 = 69\)
4Step 4: Construct the derivation tree
Construct the derivation tree as follows:
- Start with the root node that represents the final evaluated result: \(69\)
- Add two child nodes for the addition operation: \(5\) and \(64\)
- The node for \(64\) should be, in turn, the root of another subtree representing the up-arrow operation.
- Add two more child nodes for this up-arrow notation subtree: \(4\) and \(3\)
The derivation tree should look like this:
69
/ \
5 64
/ \
4 3
Key Concepts
Operation PrecedenceUp-arrow NotationExponentiation
Operation Precedence
Understanding operation precedence is crucial when it comes to solving mathematical expressions. It dictates the order in which different operations in an expression are performed. Usually, operations are performed from highest to lowest precedence, and there are widely accepted rules to follow. In general, exponentiation is performed first, followed by multiplication and division, and finally addition and subtraction.
In the context of the given exercise, where we are asked to construct a derivation tree for the expression \(5+(4 \uparrow 3)\), precedence must be carefully observed. The up-arrow notation, indicative of exponentiation, takes priority over addition. This means we evaluate \(4 \uparrow 3\) before adding the result to 5.
By respecting the order dictated by operation precedence, we prevent common calculation mistakes, and it aids in structuring the derivation tree correctly. The tree showcases this hierarchy, with the root node being the overall result, and each subsequent level representing the operation based on its precedence.
In the context of the given exercise, where we are asked to construct a derivation tree for the expression \(5+(4 \uparrow 3)\), precedence must be carefully observed. The up-arrow notation, indicative of exponentiation, takes priority over addition. This means we evaluate \(4 \uparrow 3\) before adding the result to 5.
By respecting the order dictated by operation precedence, we prevent common calculation mistakes, and it aids in structuring the derivation tree correctly. The tree showcases this hierarchy, with the root node being the overall result, and each subsequent level representing the operation based on its precedence.
Up-arrow Notation
The up-arrow notation in mathematics often signifies an operation known as exponentiation. This notation is less common than the standard caret symbol (\(^\)), but it serves the same purpose; it raises a base number to the power of an exponent.
For example, in the expression \(4 \uparrow 3\), the base is 4, and the exponent is 3. Thus, the operation involves multiplying the base number by itself as many times as the value of the exponent: \[4^3 = 4 \times 4 \times 4 = 64\].
The reason for introducing such notations could be primarily educational, allowing students to familiarize themselves with different expressions of the same operation. When constructing a derivation tree, the up-arrow notation clearly indicates that the particular operation is at the heart of the subtree representing exponentiation. This matches with it having a higher precedence over addition or subtraction.
For example, in the expression \(4 \uparrow 3\), the base is 4, and the exponent is 3. Thus, the operation involves multiplying the base number by itself as many times as the value of the exponent: \[4^3 = 4 \times 4 \times 4 = 64\].
The reason for introducing such notations could be primarily educational, allowing students to familiarize themselves with different expressions of the same operation. When constructing a derivation tree, the up-arrow notation clearly indicates that the particular operation is at the heart of the subtree representing exponentiation. This matches with it having a higher precedence over addition or subtraction.
Exponentiation
Exponentiation is a mathematical operation involving two numbers, the base 'b' and the exponent 'n'. When 'n' is a positive integer, exponentiation corresponds to repeated multiplication of the base: \(b^n\) equals the base 'b' multiplied by itself 'n-1' times. In essence, \(b^n\) can be broken down into \(b \times b \times ... \times b\) ('n' times).
In the exercise provided, the exponentiation part of the expression is initially represented through \(4 \uparrow 3\), which, when broken down, becomes 64. This part of the expression lays the foundation for the subtree of the derivation tree.
Exponentiation isn't limited to positive integers, and the concept can extend to other forms like fractional exponents, which represent roots, and negative exponents, which represent reciprocals. However, for the purposes of elementary operations and the construction of derivation trees within such exercises, we focus on the basic and most common representation.
In the exercise provided, the exponentiation part of the expression is initially represented through \(4 \uparrow 3\), which, when broken down, becomes 64. This part of the expression lays the foundation for the subtree of the derivation tree.
Exponentiation isn't limited to positive integers, and the concept can extend to other forms like fractional exponents, which represent roots, and negative exponents, which represent reciprocals. However, for the purposes of elementary operations and the construction of derivation trees within such exercises, we focus on the basic and most common representation.
Other exercises in this chapter
Problem 57
Prove each, where \(A, B,\) and \(C\) are arbitrary languages over \(\Sigma\) and \(x \in \Sigma\). If \(A \subseteq B,\) then \(A^{n} \subseteq B^{n}\) for eve
View solution Problem 57
The production rules of a grammar for simple arithmetic expressions are: $$\langle\text { expression }\rangle :=\langle\text { digit })(\langle\text { expressio
View solution Problem 58
Prove each, where \(A, B,\) and \(C\) are arbitrary languages over \(\Sigma\) and \(x \in \Sigma\). If \(A \subseteq B,\) then \(A^{*} \subseteq B^{*}\)
View solution Problem 58
The production rules of a grammar for simple arithmetic expressions are: $$\langle\text { expression }\rangle :=\langle\text { digit })(\langle\text { expressio
View solution