Mathematical Logic: Conjunction, Disjunction and Negation

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Mathematical logic in arithmetic is referred to as the study of logic.

Logic refers to the formal study of the principles of valid reasoning.
Reasoning can be of different types, including legal opinions as well as mathematical confirmations.
Certain logic can be also applied in Mathematics.

Basic mathematical logics are

Conjunction
Disjunction
Negation

A truth table is a breakdown of all the possible truth values returned by a logical expression.

Table of Content

Mathematical Logic Classification
Logical Operators in Mathematics
Formulas for Mathematical Logic
Conditional Statements
Things to Remember
Sample Questions

Key Terms: Mathematical logic, AND, OR, NOT, Conjunction, Disjunction, Negation, Set theory, Truth table, Model theory

Mathematical Logic Classification

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The field of mathematical logic can be divided into four subfields, as follows:

Set Theory
Recursion Theory
Model Theory
Proof Theory

Logical Operators In Mathematics

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The three basic logical operators used in Mathematics are:

Conjunction or (AND)
Disjunction or (OR)
Negation or (NOT)

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Formulas For Mathematical Logic

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The formulas for mathematical logic are discussed below

Conjunction or AND

The “AND” operator can be used to connect two statements.

It is also referred to as a conjunction.
“ ∧ “ is its symbolic form.
If any of the statements in this operator is untrue, the result will be false.
If both statements are true, the outcome will also be true.
There are two or more inputs but only one output on this device.

Example: Put the following simple assertions together to make a conjunction:

p: Rohan is a boy.

q: Neerja is a girl.

Answer: The conjunction of the statement p and q is given by

p ∧ q : Rohan is a boy and Neerja is a girl.

Read More: Heights and Distances

Conjunction (AND) Operator Truth Table

A (INPUT)	B (INPUT)	A AND B or A ∧ B (OUTPUT)
True	True	True
True	False	False
False	True	False
False	False	False

Read More: Value of Cos 60º

Disjunction (OR)

Using the OR operand, one can quickly connect two statements.

It is also referred to as Disjunction.
It can be expressed symbolically as “ V ”.
If any of the claims in this operator is true, then the outcome is true.
If both statements are false, the result will also be false.
There are two or more inputs, but there is only one output.

Example: Put the following simple assertions together to make a disjunction:

p: The sun shines.

q: It rains.

Answer: The disjunction of the statements p and q is given by

p ∨ q: The sun shines or it rains.

Read Further: Conic Sections

Disjunction (OR) Operator Truth Table

A (INPUT)	B (INPUT)	A OR B or A V B (OUTPUT)
True	True	True
True	False	True
False	True	True
False	False	False

Read Also: Tautology in Maths

Negation or (NOT)

Negation is a mathematical operator that returns the inverse of the provided expression.

It is also known as NOT and is indicated by the letter “ ∼ ”.
It's a procedure that yields the opposite consequence.
The output will be false if the input is true.
If the input is false, the result will be true.
It only has a single input and output.

Example: Write the negation of the statement

p: Pune is a city.

Answer: The negation of p is given by

~ p: Pune is not a city or

~ p: It is not the case that Pune is a city. or

~ p: It is false that Pune is a city.

Read More: Cosine Rule

Negation (NOT) Truth Table

A (INPUT)	NEGATION A (∼ A) (OUTPUT)
True	False
False	True

Conditional Statements

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If p and q are any two statements, the compound statement "if p then q" is termed a conditional statement or an implication and is expressed in symbolic form as

p → q or p ⇒ q.

The conditional statement (p ⇒ q) has p as the hypothesis (or antecedent) and q as the conclusion (or consequent).

Example: Express in English, the statement p → q, where

p : it is raining today

q : 2 + 3 > 4

Answer: The required conditional statement is “If it is raining today, then 2 + 3 > 4”

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Things To Remember

The study of logic is concerned with the way of reasoning.
It provides the guidelines for calculating the correctness of a particular argument in the context of theorem proof.
A statement's truth values are represented by the symbols T and F, respectively, and can be either "true" or "false."
A truth table is a list of the truth values of the resulting statements for every conceivable value assignment to the variables in a compound statement.
The number of rows is proportional to the number of statements.
The conjunction of p and q is a compound sentence made up of two simple phrases p and q joined by the connective ‘and.'
It is represented by “p ∧ q”.
The disjunction of p and q is a compound sentence made up of two simple sentences p and q connected by the connectives ‘or.'
It is represented by “p v q”.
Negation of a given assertion is a statement that is generated by modifying the truth value of a given statement by using words like "no" or "not.".
The conditional sentence of p and q is denoted by p ⇒ q and consists of two simple sentences p and q connected by the phase, if and then.

Also, Read Further:

Logarithms

Logarithm Formula

Value of Log 1

Sample Questions

Ques. Translate the following statement into symbolic form “Jake and James went up the hill.” (2 marks)

Ans. The given statement can be rewritten as “Jake went up the hill and James went up the hill”

Let p: Jake went up the hill and

q: James went up the hill.

The symbolic form of the given statement is p ∧ q.

Ques. Write the truth value of each of the following four statements: (4 marks)
i. Mumbai is in India and 2 + 3 = 6.
ii. Mumbai is in India and 2 + 3 = 5.
iii. Mumbai is in Canada and 2 + 3 = 5.
iv. Mumbai is in Canada and 2 + 3 = 6.

Ans.

has the truth value F as the truth value of the statement “2 + 3 = 6” is F.
has the truth value T as both the statement “Mumbai is in India” and “2 + 3 = 5” has the truth value T.
has truth value F as “Mumbai is in Canada” is F.
has truth value F as both the statements “Mumbai is in Canada” and “2 + 3 = 6” are false.

Ques. Write the truth value of each of the following statements: (4 marks)
i. India is in Asia or 2 + 2 = 4.
ii. India is in Asia or 2 + 4 = 5.
iii. India is in Australia or 2 + 2 = 4.
iv. India is in Australia or 2 + 4 = 5.

Ans. Only the last statement has the truth value F as both the sub-statements “India is in Australia” and “2 + 4 = 5” have the truth value F.

In (i), Both the statements “India is in Asia” and “2+2=4” are T, therefore the truth value is T.

In (iii), “India is in Australia” is False, however, “2+2=4” is T, therefore the true value of the statement is T.

Ques. Write the truth value of the negation of each of the following statements: (3 marks)
i. p: Every square is a rectangle.
ii. q: Mars is a star.
iii. r : 2 + 3 < 4

Ans.

~p is “Every square is not a rectangle” which isn't true, therefore the truth value of the following statement is F.
~q is “Mars is not a star” which is true, therefore the truth value of the following statement is T.
~ r is “ it isn't true that 2 + 3 < 4 “ which is true, therefore the truth value of the following statement is T.

Ques. Write the negation of each of the following conjunctions: (4 marks)
a. New York is in America and Wales is in England.
b. 3 + 3 = 6 and 5 < 9.

Ans.

p: New York is in America and q: Wales is in England.

Then, the conjunction in (a) is given by p ∧ q.

Now ~ p: New York is in not America, and

~ q: Wales Is not in England.

Therefore, the negation of p ∧ q is given by

~ ( p ∧ q) = New York is not America or Wales is not in England.

p : 3 + 3 = 6 and q : 5 < 9.

Then the conjunction in (b) is given by p ∧ q.

Now ~ p : 3 + 3 ≠ 6 and ~ q : 5

Then, the negation of p ∧ q is given by

~( p ∧ q) = (3 + 3 ≠ 6) or (5 > 9)

Ques. Write the negation of each of the following disjunctions: (4 marks)
a. Rohan is in Class XI or Amisha is in Class XII.
b. 8 is greater than 4 or 5 is less than 7.

Ans.

Let p : Rohan is in Class XI and

q: Amisha is in Class XII.

Then the disjunction in (a) is given by p ∨ q.

Now ~ p: Rohan is not in Class XI.

~ q: Amisha is not in Class XII.

Negation of p ∨ q is given by

~ (p ∨ q): Ram is not in Class X and Rahim is not in Class XII.

Let p: 8 is greater than 4, and q: 5 is less than 7. Then, the negation of p ∨ q is given by

~ (p ∨ q): 8 is not greater than 4 and 5 is not less than 7.

Ques. Find the component statements of the following compound statements and check whether they are true or false: (3 marks)
i. Number 3 is prime or it is odd.
ii. All integers are positive or negative.
iii. 100 is divisible by 3, 11 and 5.

Ans.

The component statements are:

p: Number 3 is the prime number.

q: It is an odd number.

Both the component statements p and q are true.

The component statements are:

p: All integers are positive.

q: All integers are negative.

Both the component statements p and q are false.

The component statements are:

p: 100 is divisible by 3.

q:100 is divisible by 11.

r: 100 is divisible by 5.

The component statements p and q are false whereas r is true.

Ques. Check whether the following pair of statements are negations of each other. Give reasons for your answer. (2 marks)
i. x+ y = y + x is true for every real numbers x and y
ii. There exists real numbers x and y for which x + y = y + x

Ans. Let p : x + y = y + x is true for every real number x and y.

And q : There exists real numbers x and y for which x + y = y + x

Now ~p: There exists real numbers and for which x + y =/ y + x therefore ~=/q.

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Mathematical Logic: Conjunction, Disjunction and Negation

Mathematical Logic Classification

Logical Operators In Mathematics

Formulas For Mathematical Logic

Conjunction or AND

Conjunction (AND) Operator Truth Table

Disjunction (OR)

Disjunction (OR) Operator Truth Table

Negation or (NOT)

Negation (NOT) Truth Table

Conditional Statements

Things To Remember

Sample Questions

CBSE CLASS XII Related Questions

1.
If A'= \(\begin{bmatrix} 3 & 4 \\ -1 & 2 \\ 0 &1 \end{bmatrix}\)\(\begin{bmatrix} -1 & 2 & 1 \\ 1 &2 & 3\end{bmatrix}\) , then verify that
(i) \((A+B)'=A'+B' \)
(ii) \((A-B)'=A'-B'\)

2.
Find the vector and the cartesian equations of the lines that pass through the origin and(5,-2,3).

3.
Find the inverse of each of the matrices,if it exists. \(\begin{bmatrix} 2 & 3\\ 5 & 7 \end{bmatrix}\)

4.
Let A=\(\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}\),show that(aI+bA)ⁿ=aⁿI+na^n-1bA,where I is the identity matrix of order 2 and n∈N

5.
Let f: R→R be defined as f(x) = 3x. Choose the correct answer.

6.
Evaluate \(\begin{vmatrix} cos\alpha cos\beta &cos\alpha sin\beta &-sin\alpha \\ -sin\beta&cos\beta &0 \\ sin\alpha cos\beta&sin\alpha\sin\beta &cos\alpha \end{vmatrix}\)

Comments

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Mathematical Logic: Conjunction, Disjunction and Negation

Mathematical Logic Classification

Logical Operators In Mathematics

Formulas For Mathematical Logic

Conjunction or AND

Conjunction (AND) Operator Truth Table

Disjunction (OR)

Disjunction (OR) Operator Truth Table

Negation or (NOT)

Negation (NOT) Truth Table

Conditional Statements

Things To Remember

Sample Questions

CBSE CLASS XII Related Questions

1. If A'= \(\begin{bmatrix} 3 & 4 \\ -1 & 2 \\ 0 &1 \end{bmatrix}\)\(\begin{bmatrix} -1 & 2 & 1 \\ 1 &2 & 3\end{bmatrix}\) , then verify that (i) \((A+B)'=A'+B' \)(ii) \((A-B)'=A'-B'\)

2. Find the vector and the cartesian equations of the lines that pass through the origin and(5,-2,3).

3. Find the inverse of each of the matrices,if it exists. \(\begin{bmatrix} 2 & 3\\ 5 & 7 \end{bmatrix}\)

4. Let A=\(\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}\),show that(aI+bA)n=anI+nan-1bA,where I is the identity matrix of order 2 and n∈N

5. Let f: R→R be defined as f(x) = 3x. Choose the correct answer.

6. Evaluate \(\begin{vmatrix} cos\alpha cos\beta &cos\alpha sin\beta &-sin\alpha \\ -sin\beta&cos\beta &0 \\ sin\alpha cos\beta&sin\alpha\sin\beta &cos\alpha \end{vmatrix}\)

Comments

SUBSCRIBE TO OUR NEWS LETTER

1.
If A'= \(\begin{bmatrix} 3 & 4 \\ -1 & 2 \\ 0 &1 \end{bmatrix}\)\(\begin{bmatrix} -1 & 2 & 1 \\ 1 &2 & 3\end{bmatrix}\) , then verify that
(i) \((A+B)'=A'+B' \)
(ii) \((A-B)'=A'-B'\)

2.
Find the vector and the cartesian equations of the lines that pass through the origin and(5,-2,3).

3.
Find the inverse of each of the matrices,if it exists. \(\begin{bmatrix} 2 & 3\\ 5 & 7 \end{bmatrix}\)

4.
Let A=\(\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}\),show that(aI+bA)ⁿ=aⁿI+na^n-1bA,where I is the identity matrix of order 2 and n∈N

5.
Let f: R→R be defined as f(x) = 3x. Choose the correct answer.

6.
Evaluate \(\begin{vmatrix} cos\alpha cos\beta &cos\alpha sin\beta &-sin\alpha \\ -sin\beta&cos\beta &0 \\ sin\alpha cos\beta&sin\alpha\sin\beta &cos\alpha \end{vmatrix}\)