Definitions [6]
Definition: Simple and Compound Statements
A statement which cannot be split further into two or more statements is called a simple statement. If a statement is the combination of two or more simple statements, then it is called a compound statement.
Definition: Logical Equivalence
Two statement patterns are said to be equivalent if their truth tables are identical. If statement patterns A and B are equivalent, we write it as A ≡ B.
Definition: Statement
A statement is a declarative (assertive) sentence which is either true or false, but not both simultaneously. Statements are denoted by p, q, r, ....
Open Statement:
An open sentence is a sentence whose truth can vary depending on conditions not stated in the sentence.
Definition: Truth Value of a Statement
Each statement is either true or false. If a statement is true, then its truth value is 'T', and if the statement is false, then its truth value is F.
Definition: Logical Connectives
The words or phrases which are used to connect two statements are called logical connectives.
Eg. 'and', 'or', 'if ..... then', 'if and only if ', 'not''.
Definition: Duality
Two compound statements s1 and s2 are said to be duals of each other if one can be obtained from the other by:
-
Replacing ∧ (and) by ∨ (or)
-
Replacing ∨ (or) by ∧ (and)
-
Replacing T (tautology) by F (contradiction)
-
Replacing F (contradiction) by T (tautology)
while keeping negations unchanged.
Theorems and Laws [1]
Using truth table prove that ~ p ˄ q ≡ ( p ˅ q) ˄ ~ p
~ p ∧ q ≡ (p ∨ q) ∧ ~ p
| 1 | 2 | 3 | 4 | 5 | 6 |
| p | q | ~p | ~p ∧ q | (p ∨ q) | (p ∨ q) ∧ ~p |
| T | T | F | F | T | F |
| T | F | F | F | T | F |
| F | T | T | T | T | T |
| F | F | T | F | F | F |
In the above truth table, the entries in the columns 4 and 6 are identical.
∴ ~p ∧ q ≡ (p ∨ q) ∧ ~p.
Key Points
Key Points: Venn Diagram
| Logical Form | Set Form |
|---|---|
| p ∧ q | A ∩ B |
| p ∨ q | A ∪ B |
| ∼p | A′ (complement) |
| p → q | A ⊂ B |
| p ↔ q | A = B |
Key Points: Tautology, Contradiction and Contingency
| Type | Meaning |
|---|---|
| Tautology | Always True |
| Contradiction | Always False |
| Contingency | Sometimes T, Sometimes F |
Key Points: Quantifiers and Quantified Statements
| Symbol | Meaning |
|---|---|
| ∀ | For all (Universal quantifier) |
| ∃ | There exists (Existential quantifier) |
Quantified statement: An open sentence with a quantifier becomes a statement and is called a quantified statement.
Key Points: Switching Circuits
| Circuit Type | Logical Form |
|---|---|
| Series | p ∧ q |
| Parallel | p ∨ q |
Switch ON = 1
Switch OFF = 0
Key Points: Converse, Inverse, Contrapositive
For p → q:
| Type | Form |
|---|---|
| Converse | q → p |
| Inverse | ∼p → ∼q |
| Contrapositive | ∼q → ∼p |
Key Points: Negation of Compound Statements
| Type | Given Statement | Negation | Symbolic Form |
|---|---|---|---|
| Negation of Conjunction | p ∧ q | Not p or Not q | ~(p ∧ q) ≡ ~p ∨ ~q |
| Negation of Disjunction | p ∨ q | Not p and Not q | ~(p ∨ q) ≡ ~p ∧ ~q |
| Negation of Implication | p → q | p and Not q | ~(p → q) ≡ p ∧ ~q |
| Negation of Biconditional | p ↔ q | (p and Not q) or (q and Not p) | ~(p ↔ q) ≡ (p ∧ ~q) ∨ (q ∧ ~p) |
| Negation of Quantified Statement | ∀ x P(x) / ∃ x P(x) | Replace “all” by “some” and vice versa, and negate P(x) | ~(∀ x P(x)) ≡ ∃x ~P(x) ~(∃x P(x)) ≡ ∀x ~P(x) |
Key Points: Algebra of Statements
| Sr. No. | Law Name | Logical Form |
|---|---|---|
| 1 | Idempotent Law | p ∧ p ≡ p p ∨ p ≡ p |
| 2 | Commutative Law | p ∧ q ≡ q ∧ p p ∨ q ≡ q ∨ p |
| 3 | Associative Law | p ∧ (q ∧ r) ≡ (p ∧ q) ∧ r ≡ p ∧ q ∧ r p ∨ (q ∨ r) ≡ (p ∨ q) ∨ r ≡ p ∨ q ∨ r |
| 4 | Distributive Law | p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r) p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r) |
| 5 | De Morgan’s Laws | ~(p ∧ q) ≡ ~p ∨ ~q ~(p ∨ q) ≡ ~p ∧ ~q |
| 6 | Identity Laws | p ∧ T ≡ p p ∨ F ≡ p p ∧ F ≡ F p ∨ T ≡ T |
| 7 | Complement Laws | p ∧ ~p ≡ F p ∨ ~p ≡ T |
| 8 | Absorption Laws | p ∨ (p ∧ q) ≡ p p ∧ (p ∨ q) ≡ p |
| 9 |
Conditional Law |
p → q ≡ ~p ∨ q |
| 10 | Biconditional Law |
p ↔ q ≡ (p → q) ∧ (q → p) ≡ (~p ∨ q) ∧ (~q ∨ p) |
Key Points: Logical Connectives
| Connective | Symbol | Name | True When |
|---|---|---|---|
| and | ∧ | Conjunction | Both true |
| or | ∨ | Disjunction | At least one true |
| if...then | → | Conditional | False only when T → F |
| iff | ↔ | Biconditional | Same truth values |
| not | ~ | Negation | Opposite value |
Note: ~ (~ p) = p
Important Questions [52]
- The dual of the statement (p ˅ q) ˄ (r ˅ s) is ______.
- Which of the following is not a statement?
- The negation of the proposition “If 2 is prime, then 3 is odd”, is ______.
- Write the converse, inverse, and contrapositive of the statement. "If 2 + 5 = 10, then 4 + 10 = 20."
- If p : He swims q : Water is warm Give the verbal statement for the following symbolic statement. ~ (p ∨ q)
- If p : He swims q : Water is warm Give the verbal statement for the following symbolic statement: p ↔ ~ q
- Write Converse, Inverse Contrnpositlve of the Statement "If Two Bi Act Arc Not Congruent Then Their Areas Are Not Equal.
- Evaluate: ∫ X . Log X Dx
- If p : He swims q : Water is warm Give the verbal statement for the following symbolic statement. q → p
- Negation of “Some men are animal” is ______.
- Consider the following statements. If D is dog, then D is very good. If D is very good, then D is dog. If D is not very good, then D is not a dog.
- Converse of the statement q → p is ______.
- State the following sentence is statement. In case of statement, write down the truth value : Every quadratic equation has only real roots.
- State the Following Sentence is Statement. in Case of Statement, Write Down the Truth Value : √ -4 is a Rational Number.
- Write Converse and Inverse of the Following Statement : If Ravi is Good in Logic Then Ravi is Good in Mathematics
- By Constructing the Truth Table, Deterdline Whether the Following Statement Pattern Ls a Tautology,
- Examine Whether the Following Statement (P ∧ Q) ∨ (∼P ∨ ∼Q) is a Tautology Or Contradiction Or Neither of Them.
- Using the truth table, prove the following logical equivalence. p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)
- Write the negation of the following statement: (p → q) ∨ (p → r)
- Prove that the following statement pattern is equivalent : (p ∨ q) r and (p → r) ∧ (q → r)
- Write Converse and Inverse of the Following Statement: “If a Man is a Bachelor Then He is Unhappy.”
- Prove that the Following Statement Pattern is a Tautology : ( Q → P ) V ( P → Q )
- If P and Q Are True Statements and R and S Are False Statements, Find the Truth Value of the Following : ( P ∧ ∼ R ) ∧ ( ∼ Q ∧ S)
- If P : It is Raining Q : It is Humid Write the Following Statements in Symbolic Form: (A) It is Raining Or Humid.
- Using Truth Table, Examine Whether the Following Statement Pattern is Tautology, Contradiction Or Contingency: P ∨ ∼(P ∧ Q)
- Show that the Following Statement Pattern in Contingency :
- Use the Quantifiers to Convert the Following Open Sentence Defined on N into True Statement 5x - 3 < 10
- Use the Quantifiers to Convert the Following Open Sentence Defined on N into True Statement: X2 ≥ 1
- Write the Negation of the Following Statement : ∀ Y ∈ N, Y2 + 3 ≤ 7
- Write the Negation of the Following Statement : If the Lines Are Parallel Then Their Slopes Are Equal.
- Examine whether the following statement pattern is a tautology, a contradiction or a contingency. (p ∧ ~ q) → (~ p ∧ ~ q)
- Examine whether the following statement pattern is a tautology, a contradiction or a contingency. ~ p → (p → ~ q)
- Using the truth table, verify. p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)
- Using the truth table, verify ~(p → ~q) ≡ p ∧ ~ (~ q) ≡ p ∧ q.
- Write the negation of the following statement. ∃ n ∈ N, (n2 + 2) is odd number.
- Write the negation of the following statement. Some continuous functions are differentiable.
- Construct the truth table for the following statement pattern. (p ∧ ~ q) ↔ (q → p)
- Determine whether the following statement pattern is a tautology, contradiction, or contingency. [(p ∧ q) ∨ (~p)] ∨ [p ∧ (~ q)]
- Write the converse, inverse, contrapositive of the following statement. If a man is bachelor, then he is happy.
- Determine whether the following statement pattern is a tautology, contradiction or contingency: [(∼ p ∧ q) ∧ (q ∧ r)] ∧ (∼ q)
- Express the following statement in symbolic form and write its truth value. "If 4 is an odd number, then 6 is divisible by 3"
- If p ∨ q is true, then the truth value of ∼ p ∧ ∼ q is ______.
- Using the Rules of Negation, Write the Negatlon of the Following:
- Write the Truth Value of the Negation of the Following Statement :
- Write the Truth Value of the Negation of the Following Statement :
- Express the Truth of Each of the Following Statements by Venn Diagram: (A) Some Hardworking Students Are Obedient. (B) No Circles Are Polygons. (C) All Teachers Are Scholars and Scholars Are Teachers.
- Express the Truth of Each of the Following Statements Using Venn Diagrams:
- Express the Truth of Each of the Following Statements Using Venn Diagram. (1) All Teachers Are Scholars and Scholars Are Teachers. (2) If a Quadrilateral is a Rhombus Then It is a Parallelogram..
- Draw Venn Diagram for the Truth of the Following Statements : Some Rectangles Are Squares.
- Using the Venn Diagram. Examine the Logical Equivalence of the Following Statements:
- Draw Venn Diagram for the Truth of the Following Statements : All Rational Number Are Real Numbers.
- Express the Truth of the Following Statements with the Help of Venn Diagram: (A) No Circles Are Polygon (B) If a Quadrilateral is Rhombus , Then It is a Parallelogram .
