Elementary Logic
It’s the study of processes used in mathematical induction.
There are 2 types of reasoning:
Inductive → conclusions based on observations.
Deductive → conclusions based on assumption.
proof: argument that is sufficient to convince a person about the validity of a certain result.
the study of logical relationships between propositions.
A proposition is a statement that can be either true or false, but not both.
For example,
"It is raining" and "2 + 2 = 4" are propositions.( true logical statement)
Pluto is a planet (false statement)
statement like ; x+1=1 is not a logical statement
Truth value: the truth or falsity of a statement /proposition
Propositional logic uses logical operators to combine or modify propositions. The main logical operators are:
Negation (¬):
Denotes the logical opposite of a proposition(logical statement)
If p is a proposition, then ¬p is true when p is false, and false when p is true.
Apply the logical NOT method.
example: p =The bottle is full.
negation: ¬p =the bottle is not full./ empty
Conjunction (∧):
propositions joined by the logical connective ‘and’.
A proposition p^q is true when both p and q are true simultaneously
t=1 f=0
Apply the logical “AND” method. :i.e multiplication)
Truth table:
p | q | p^q |
|---|---|---|
T | T | T |
T | F | F |
F | T | F |
F | F | F |
Disjunction (∨):
Categorized into 2:
i) Inclusive(∨):
pvq is true if at least one of them is true .
If both p and q are false pvq is false.
Apply Logical OR i.e addition.
Truth table:
p | q | pvq |
|---|---|---|
T | T | T |
T | F | T |
F | T | T |
F | F | F |
ii)Exclusive(⊻):
If both p and q have the same truth value p⊻q is false.
Apply Logical XOR.
Truth table:
p | q | p⊻q |
|---|---|---|
T | T | F |
T | F | T |
F | T | T |
F | F | F |
Implication/Conditional (→):
propositions joined by the logical connective ‘if…then’.
A proposition p→q is true when both p and q are true and when p is false.
false when only P is true
p: It is raining
q:It is wet.
Truth table:
p | q | p→q |
|---|---|---|
T | T | T |
T | F | F |
F | T | T |
F | F | T |
Biconditional (↔):
propositions joined by the logical connective ‘if and only if’.
A proposition p↔q is true when both p and q have the same truth value
Truth table:
p | q | p↔q |
|---|---|---|
T | T | T |
T | F | F |
F | T | F |
F | F | T |
A tautology is a statement that is always true, regardless of the truth values of its individual components.
Example:
p: It’s raining outside; then pv~p is Its raining outside or its not raining outside = True
truth table:
p | ~p | p v ~p |
|---|---|---|
T | F | T |
F | T | T |
A contradiction is a statement that is always false, regardless of the truth values of its components.
Example:
A: It is raining. so A ^ ~A(It’s raining and it’s not raining) = F since it can’t rain and not rain at the same time.
Truth table
A | ~A | A ^ ~A |
|---|---|---|
T | F | F |
F | T | F |
Two statements are said to be logically equivalent if they always have the same truth value, regardless of the truth values of their components.
Symbolically, two statements P and Q are logically equivalent if and only if P ↔ Q is a tautology.
example: prove p is logically equivalent to p^T
p | T | p^T |
|---|---|---|
T | T | T |
F | T | F |
The relationship between the converse, inverse, and contrapositive of a conditional statement is as follows:
P: Jack plays his guitar
Q:Sarah will sing.
Converse: The converse of a conditional statement switches the hypothesis and conclusion. If the original statement is "If P, then Q," the converse would be "If Q, then P." However, the converse may or may not be true. Just because the original statement is true does not guarantee the truth of its converse. P →Q
Inverse: The inverse of a conditional statement negates both the hypothesis and the conclusion. If the original statement is "If P, then Q," the inverse would be "If not P, then not Q." Similar to the converse, the inverse may or may not be true. The truth of the original statement does not imply the truth of its inverse.
Contrapositive: The contrapositive of a conditional statement combines both the switching of hypothesis and conclusion (like the converse) and the negation of both (like the inverse). If the original statement is "If P, then Q," the contrapositive would be "If not Q, then not P." Unlike the converse and inverse, the contrapositive is always true if the original statement is true. If the original statement is false, the contrapositive is also false.
Logic is a fundamental tool used in mathematical proof to establish the validity of mathematical statements.
Mathematical proof involves a systematic and rigorous process of reasoning to demonstrate the truth of a mathematical statement.
method used to show that a statement or claim is false by providing just one example that contradicts it.
a technique used to disprove a statement or conjecture by providing a specific example that contradicts it.
It is a powerful tool that allows us to show that a general statement is not always true
example:
Statement: "All prime numbers are odd."
To disprove this statement, we need to find a counterexample, i.e., a prime number that is not odd.
e,g, the number 2 is a counter example because it is a prime number but it is not odd.
The number 2 is a prime number, but it is not odd. Therefore, the statement "All prime numbers are odd" is false.
A direct proof is a method of demonstrating the truth of a statement (to show the validity of a given statement) by logically connecting the known facts
It involves a logical sequence of steps that lead from known facts or assumptions to the desired conclusion. In a direct proof, we start with the given information and apply logical reasoning to reach the desired result.
A direct proof typically consists of the following components:
Statement of the Given: Begin by stating the given information or assumptions that are provided in the problem.
Statement of the Goal: Clearly state the proposition or statement that you are trying to prove.
Logical Reasoning: Use logical reasoning, definitions, axioms, and previously proven theorems to establish a chain of logical steps leading from the given information to the desired conclusion.
Conclusion: Summarize the logical steps and state the conclusion that follows from the given information.
example:
Proposition: If a and b are even integers, then a + b is also an even integer.
Proof:
Given: a and b are even integers.
Let a = 2m and b = 2n, where m and n are integers.
Substitute the values of a and b into the expression a + b:
a + b = 2m + 2n
a + b = 2(m + n)
Since m + n is an integer (sum of two integers), a + b can be expressed as 2k, where k = m + n.
Therefore, a + b is an even integer.
In this example, we started with the given information that a and b are even integers. By using the definition of even integers and performing algebraic manipulations, we established that a + b can be expressed as 2k, where k is an integer. Hence, we concluded that a + b is an even integer.
Proof by cases is a technique used in mathematics to establish the truth of a statement by considering different possible scenarios or cases. It involves breaking down the problem into distinct cases and proving the statement for each case individually. Here's how it works:
Statement: Start with the statement you want to prove. For example, let's consider the statement: "For all integers n, if n is even, then n^2 is also even."
Cases: Identify the different cases that need to be considered. In this example, we have two cases: when n is an even integer and when n is an odd integer.
Proof for each case: For each case, provide a separate proof to establish the truth of the statement.
Case 1: n is even: Assume n is an even integer. Then, n can be expressed as n = 2k, where k is an integer. Now, let's consider n^2 = (2k)^2 = 4k^2. Since 4k^2 is divisible by 2, n^2 is also even. Thus, the statement holds true for this case.
Case 2: n is odd: Assume n is an odd integer. Then, n can be expressed as n = 2k + 1, where k is an integer. Now, let's consider n^2 = (2k + 1)^2 = 4k^2 + 4k + 1. Notice that 4k^2 + 4k is divisible by 2, and adding 1 to an even number results in an odd number. Therefore, n^2 is odd. Thus, the statement holds true for this case as well.
Conclusion: Since we have proven the statement for both cases, we can conclude that the statement "For all integers n, if n is even, then n^2 is also even" is true.
Proof by cases allows us to handle situations where a single approach may not be sufficient to prove a statement for all possible scenarios. By breaking down the problem into cases and providing individual proofs, we can establish the validity of the statement as a whole.
Example: Prove that for any real number x, if x > 0, then x^2 > 0.
Proof by cases:
Case 1: x > 1
To prove the statement for this case, let's assume that x is greater than 1. Since x is positive, we know that x^2 will also be positive. This is because when we square a positive number, the result is always positive. Therefore, for this case, x^2 is greater than 0.
Case 2: 0 < x < 1
In this case, let's assume that x is a positive number between 0 and 1. When we square a number between 0 and 1, the result is smaller than the original number. For example, if we take x = 0.5, then x^2 = 0.25, which is smaller than 0.5. Therefore, for this case as well, x^2 is greater than 0.
Case 3: x = 0
When x is equal to 0, x^2 will also be equal to 0. Since 0 is neither greater nor smaller than 0, we can say that x^2 is equal to 0. Therefore, for this case, x^2 is greater than 0.
Now, we have considered all possible cases and have proven that for each case, x^2 is greater than 0. Hence, we can conclude that for any real number x, if x > 0, then x^2 > 0.
Proof by cases is a powerful technique that allows us to handle different scenarios and establish the truth of a statement in a systematic manner. It helps in breaking down complex problems into manageable cases and providing separate proofs for each case. By considering all possible cases, we can ensure that the statement holds true for all scenarios.
Proof by mathematical induction is a powerful technique used to prove statements about natural numbers. It consists of two steps: the base case and the inductive step.
The base case is the initial step of the proof, where we verify that the statement holds true for the smallest possible value of the natural number. Typically, this is the number 1 or 0, depending on the context.
For example, let's prove that the sum of the first n natural numbers is given by the formula: 1 + 2 + 3 + ... + n = n(n+1)/2.
Base Case: When n = 1, the left-hand side of the equation becomes 1, and the right-hand side becomes (1)(1+1)/2 = 1. Thus, the formula holds true for n = 1.
The inductive step is where we assume that the statement holds true for a particular value of n, and then prove that it also holds true for the next value, n+1.
For the same example, assuming that the formula holds true for some k, we need to show that it also holds true for k+1.
Inductive Hypothesis: Assume that 1 + 2 + 3 + ... + k = k(k+1)/2.
Inductive Step: We need to prove that 1 + 2 + 3 + ... + k + (k+1) = (k+1)((k+1)+1)/2.
Adding (k+1) to both sides of the inductive hypothesis, we get:
k(k+1)/2 + (k+1) = (k+1)((k+1)+1)/2.
Simplifying the left-hand side, we have:
(k^2 + k + 2k + 2) / 2 = (k+1)(k+2)/2.
Rearranging the terms, we get:
(k^2 + 3k + 2) / 2 = (k+1)(k+2)/2.
This is equivalent to:
(k+1)(k+2)/2 = (k+1)(k+2)/2.
Thus, the formula holds true for k+1.
By proving the base case and the inductive step, we have established that the statement holds true
for all natural numbers. Therefore, the formula 1 + 2 + 3 + ... + n = n(n+1)/2 is proven by mathematical induction.
Mathematical induction is a fundamental technique in mathematics that allows us to prove statements about natural numbers. It is based on the principle that if a statement holds true for a base case and can be shown to hold true for the next value using the inductive step, then it holds true for all values of the natural numbers.
The base case is the starting point of the proof. It is where we verify that the statement holds true for the smallest possible value of the natural number. In the given example, the base case is when n = 1. By substituting n = 1 into the formula, we can see that both sides of the equation are equal to 1. Therefore, the formula holds true for the base case.
After establishing the base case, we move on to the inductive step. In this step, we assume that the statement holds true for a particular value of n, which we denote as k. This assumption is called the inductive hypothesis. Then, we need to prove that the statement also holds true for the next value, k+1.
To prove the inductive step, we start by adding (k+1) to both sides of the inductive hypothesis. This allows us to manipulate the equation and simplify it. By rearranging the terms and performing algebraic manipulations, we arrive at the conclusion that the formula holds true for k+1.
By proving the base case and the inductive step, we have established that the statement holds true for all natural numbers. This is the essence of mathematical induction. It is a powerful tool that is widely used in various branches of mathematics, such as number theory, combinatorics, and calculus.
In conclusion, proof by mathematical induction is a systematic and rigorous method for proving statements about natural numbers. It involves verifying the base case and then proving the inductive step to establish the truth of the statement for all values of the natural numbers. By following this method, we can confidently prove mathematical statements and contribute to the advancement of mathematical knowledge.
QUESTIONS
https://www.brainkart.com/article/Exercise-12-2--Mathematical-Logic_41293/
It’s the study of processes used in mathematical induction.
There are 2 types of reasoning:
Inductive → conclusions based on observations.
Deductive → conclusions based on assumption.
proof: argument that is sufficient to convince a person about the validity of a certain result.
the study of logical relationships between propositions.
A proposition is a statement that can be either true or false, but not both.
For example,
"It is raining" and "2 + 2 = 4" are propositions.( true logical statement)
Pluto is a planet (false statement)
statement like ; x+1=1 is not a logical statement
Truth value: the truth or falsity of a statement /proposition
Propositional logic uses logical operators to combine or modify propositions. The main logical operators are:
Negation (¬):
Denotes the logical opposite of a proposition(logical statement)
If p is a proposition, then ¬p is true when p is false, and false when p is true.
Apply the logical NOT method.
example: p =The bottle is full.
negation: ¬p =the bottle is not full./ empty
Conjunction (∧):
propositions joined by the logical connective ‘and’.
A proposition p^q is true when both p and q are true simultaneously
t=1 f=0
Apply the logical “AND” method. :i.e multiplication)
Truth table:
p | q | p^q |
|---|---|---|
T | T | T |
T | F | F |
F | T | F |
F | F | F |
Disjunction (∨):
Categorized into 2:
i) Inclusive(∨):
pvq is true if at least one of them is true .
If both p and q are false pvq is false.
Apply Logical OR i.e addition.
Truth table:
p | q | pvq |
|---|---|---|
T | T | T |
T | F | T |
F | T | T |
F | F | F |
ii)Exclusive(⊻):
If both p and q have the same truth value p⊻q is false.
Apply Logical XOR.
Truth table:
p | q | p⊻q |
|---|---|---|
T | T | F |
T | F | T |
F | T | T |
F | F | F |
Implication/Conditional (→):
propositions joined by the logical connective ‘if…then’.
A proposition p→q is true when both p and q are true and when p is false.
false when only P is true
p: It is raining
q:It is wet.
Truth table:
p | q | p→q |
|---|---|---|
T | T | T |
T | F | F |
F | T | T |
F | F | T |
Biconditional (↔):
propositions joined by the logical connective ‘if and only if’.
A proposition p↔q is true when both p and q have the same truth value
Truth table:
p | q | p↔q |
|---|---|---|
T | T | T |
T | F | F |
F | T | F |
F | F | T |
A tautology is a statement that is always true, regardless of the truth values of its individual components.
Example:
p: It’s raining outside; then pv~p is Its raining outside or its not raining outside = True
truth table:
p | ~p | p v ~p |
|---|---|---|
T | F | T |
F | T | T |
A contradiction is a statement that is always false, regardless of the truth values of its components.
Example:
A: It is raining. so A ^ ~A(It’s raining and it’s not raining) = F since it can’t rain and not rain at the same time.
Truth table
A | ~A | A ^ ~A |
|---|---|---|
T | F | F |
F | T | F |
Two statements are said to be logically equivalent if they always have the same truth value, regardless of the truth values of their components.
Symbolically, two statements P and Q are logically equivalent if and only if P ↔ Q is a tautology.
example: prove p is logically equivalent to p^T
p | T | p^T |
|---|---|---|
T | T | T |
F | T | F |
The relationship between the converse, inverse, and contrapositive of a conditional statement is as follows:
P: Jack plays his guitar
Q:Sarah will sing.
Converse: The converse of a conditional statement switches the hypothesis and conclusion. If the original statement is "If P, then Q," the converse would be "If Q, then P." However, the converse may or may not be true. Just because the original statement is true does not guarantee the truth of its converse. P →Q
Inverse: The inverse of a conditional statement negates both the hypothesis and the conclusion. If the original statement is "If P, then Q," the inverse would be "If not P, then not Q." Similar to the converse, the inverse may or may not be true. The truth of the original statement does not imply the truth of its inverse.
Contrapositive: The contrapositive of a conditional statement combines both the switching of hypothesis and conclusion (like the converse) and the negation of both (like the inverse). If the original statement is "If P, then Q," the contrapositive would be "If not Q, then not P." Unlike the converse and inverse, the contrapositive is always true if the original statement is true. If the original statement is false, the contrapositive is also false.
Logic is a fundamental tool used in mathematical proof to establish the validity of mathematical statements.
Mathematical proof involves a systematic and rigorous process of reasoning to demonstrate the truth of a mathematical statement.
method used to show that a statement or claim is false by providing just one example that contradicts it.
a technique used to disprove a statement or conjecture by providing a specific example that contradicts it.
It is a powerful tool that allows us to show that a general statement is not always true
example:
Statement: "All prime numbers are odd."
To disprove this statement, we need to find a counterexample, i.e., a prime number that is not odd.
e,g, the number 2 is a counter example because it is a prime number but it is not odd.
The number 2 is a prime number, but it is not odd. Therefore, the statement "All prime numbers are odd" is false.
A direct proof is a method of demonstrating the truth of a statement (to show the validity of a given statement) by logically connecting the known facts
It involves a logical sequence of steps that lead from known facts or assumptions to the desired conclusion. In a direct proof, we start with the given information and apply logical reasoning to reach the desired result.
A direct proof typically consists of the following components:
Statement of the Given: Begin by stating the given information or assumptions that are provided in the problem.
Statement of the Goal: Clearly state the proposition or statement that you are trying to prove.
Logical Reasoning: Use logical reasoning, definitions, axioms, and previously proven theorems to establish a chain of logical steps leading from the given information to the desired conclusion.
Conclusion: Summarize the logical steps and state the conclusion that follows from the given information.
example:
Proposition: If a and b are even integers, then a + b is also an even integer.
Proof:
Given: a and b are even integers.
Let a = 2m and b = 2n, where m and n are integers.
Substitute the values of a and b into the expression a + b:
a + b = 2m + 2n
a + b = 2(m + n)
Since m + n is an integer (sum of two integers), a + b can be expressed as 2k, where k = m + n.
Therefore, a + b is an even integer.
In this example, we started with the given information that a and b are even integers. By using the definition of even integers and performing algebraic manipulations, we established that a + b can be expressed as 2k, where k is an integer. Hence, we concluded that a + b is an even integer.
Proof by cases is a technique used in mathematics to establish the truth of a statement by considering different possible scenarios or cases. It involves breaking down the problem into distinct cases and proving the statement for each case individually. Here's how it works:
Statement: Start with the statement you want to prove. For example, let's consider the statement: "For all integers n, if n is even, then n^2 is also even."
Cases: Identify the different cases that need to be considered. In this example, we have two cases: when n is an even integer and when n is an odd integer.
Proof for each case: For each case, provide a separate proof to establish the truth of the statement.
Case 1: n is even: Assume n is an even integer. Then, n can be expressed as n = 2k, where k is an integer. Now, let's consider n^2 = (2k)^2 = 4k^2. Since 4k^2 is divisible by 2, n^2 is also even. Thus, the statement holds true for this case.
Case 2: n is odd: Assume n is an odd integer. Then, n can be expressed as n = 2k + 1, where k is an integer. Now, let's consider n^2 = (2k + 1)^2 = 4k^2 + 4k + 1. Notice that 4k^2 + 4k is divisible by 2, and adding 1 to an even number results in an odd number. Therefore, n^2 is odd. Thus, the statement holds true for this case as well.
Conclusion: Since we have proven the statement for both cases, we can conclude that the statement "For all integers n, if n is even, then n^2 is also even" is true.
Proof by cases allows us to handle situations where a single approach may not be sufficient to prove a statement for all possible scenarios. By breaking down the problem into cases and providing individual proofs, we can establish the validity of the statement as a whole.
Example: Prove that for any real number x, if x > 0, then x^2 > 0.
Proof by cases:
Case 1: x > 1
To prove the statement for this case, let's assume that x is greater than 1. Since x is positive, we know that x^2 will also be positive. This is because when we square a positive number, the result is always positive. Therefore, for this case, x^2 is greater than 0.
Case 2: 0 < x < 1
In this case, let's assume that x is a positive number between 0 and 1. When we square a number between 0 and 1, the result is smaller than the original number. For example, if we take x = 0.5, then x^2 = 0.25, which is smaller than 0.5. Therefore, for this case as well, x^2 is greater than 0.
Case 3: x = 0
When x is equal to 0, x^2 will also be equal to 0. Since 0 is neither greater nor smaller than 0, we can say that x^2 is equal to 0. Therefore, for this case, x^2 is greater than 0.
Now, we have considered all possible cases and have proven that for each case, x^2 is greater than 0. Hence, we can conclude that for any real number x, if x > 0, then x^2 > 0.
Proof by cases is a powerful technique that allows us to handle different scenarios and establish the truth of a statement in a systematic manner. It helps in breaking down complex problems into manageable cases and providing separate proofs for each case. By considering all possible cases, we can ensure that the statement holds true for all scenarios.
Proof by mathematical induction is a powerful technique used to prove statements about natural numbers. It consists of two steps: the base case and the inductive step.
The base case is the initial step of the proof, where we verify that the statement holds true for the smallest possible value of the natural number. Typically, this is the number 1 or 0, depending on the context.
For example, let's prove that the sum of the first n natural numbers is given by the formula: 1 + 2 + 3 + ... + n = n(n+1)/2.
Base Case: When n = 1, the left-hand side of the equation becomes 1, and the right-hand side becomes (1)(1+1)/2 = 1. Thus, the formula holds true for n = 1.
The inductive step is where we assume that the statement holds true for a particular value of n, and then prove that it also holds true for the next value, n+1.
For the same example, assuming that the formula holds true for some k, we need to show that it also holds true for k+1.
Inductive Hypothesis: Assume that 1 + 2 + 3 + ... + k = k(k+1)/2.
Inductive Step: We need to prove that 1 + 2 + 3 + ... + k + (k+1) = (k+1)((k+1)+1)/2.
Adding (k+1) to both sides of the inductive hypothesis, we get:
k(k+1)/2 + (k+1) = (k+1)((k+1)+1)/2.
Simplifying the left-hand side, we have:
(k^2 + k + 2k + 2) / 2 = (k+1)(k+2)/2.
Rearranging the terms, we get:
(k^2 + 3k + 2) / 2 = (k+1)(k+2)/2.
This is equivalent to:
(k+1)(k+2)/2 = (k+1)(k+2)/2.
Thus, the formula holds true for k+1.
By proving the base case and the inductive step, we have established that the statement holds true
for all natural numbers. Therefore, the formula 1 + 2 + 3 + ... + n = n(n+1)/2 is proven by mathematical induction.
Mathematical induction is a fundamental technique in mathematics that allows us to prove statements about natural numbers. It is based on the principle that if a statement holds true for a base case and can be shown to hold true for the next value using the inductive step, then it holds true for all values of the natural numbers.
The base case is the starting point of the proof. It is where we verify that the statement holds true for the smallest possible value of the natural number. In the given example, the base case is when n = 1. By substituting n = 1 into the formula, we can see that both sides of the equation are equal to 1. Therefore, the formula holds true for the base case.
After establishing the base case, we move on to the inductive step. In this step, we assume that the statement holds true for a particular value of n, which we denote as k. This assumption is called the inductive hypothesis. Then, we need to prove that the statement also holds true for the next value, k+1.
To prove the inductive step, we start by adding (k+1) to both sides of the inductive hypothesis. This allows us to manipulate the equation and simplify it. By rearranging the terms and performing algebraic manipulations, we arrive at the conclusion that the formula holds true for k+1.
By proving the base case and the inductive step, we have established that the statement holds true for all natural numbers. This is the essence of mathematical induction. It is a powerful tool that is widely used in various branches of mathematics, such as number theory, combinatorics, and calculus.
In conclusion, proof by mathematical induction is a systematic and rigorous method for proving statements about natural numbers. It involves verifying the base case and then proving the inductive step to establish the truth of the statement for all values of the natural numbers. By following this method, we can confidently prove mathematical statements and contribute to the advancement of mathematical knowledge.
QUESTIONS
https://www.brainkart.com/article/Exercise-12-2--Mathematical-Logic_41293/
