Mathematical Reasoning
Statements and Connectives
A statement is a sentence that is either true or false. Connectives "and" (∧), "or" (∨) and "not" (¬) build compound statements.
Example 1: Is "x + 2 = 5" a statement?
Open sentence — true/false depends on x, so not a statement by itself.
Example 2: Negate "All birds fly".
"Some birds do not fly".
Quick recap
- A statement is definitely true or false.
- Connectives: and (∧), or (∨), not (¬).
✓ Quick check
What is the negation of the statement: '√5 is an irrational number'?
The negation simply asserts that the original statement is false. '√5 is not an irrational number' is the direct negation. Since real numbers are either rational or irrational, '√5 is a rational number' is also equivalent.
How many rows are there in the truth table of two propositions?
There are 2² = 4 possible combinations.
Quantifiers and Implications
Quantifiers: "for all" (∀) and "there exists" (∃). An implication "if p then q" has converse q→p and contrapositive ¬q→¬p (logically equivalent to the original).
Example 1: Contrapositive of "if it rains, the ground is wet".
"If the ground is not wet, it did not rain".
Example 2: Converse of "if p then q".
"If q then p".
Quick recap
- Contrapositive ≡ original; converse is different.
- ∀ = for all, ∃ = there exists.
✓ Quick check
Identify the truth value of the statement: 'Every square is a rectangle.'
A square has all the properties of a rectangle (opposite sides equal and all angles 90°). Therefore, the statement is True.
Which of the following is an open sentence rather than a statement?
An open sentence contains a variable and its truth value cannot be determined until a specific value is substituted. 'x² − 5x + 6 = 0' is true for x = 2, 3 but false otherwise.
Validating Statements
Prove "and" by proving both parts; "or" by showing one holds. Disprove a "for all" claim with a single counter-example.
Example 1: Disprove "every prime is odd".
2 is prime and even — a counter-example.
Example 2: To prove "p and q", what is needed?
Prove p and prove q.
Quick recap
- One counter-example disproves a universal claim.
- For "and" prove both; for "or" prove one.
✓ Quick check
Let p be 'Delhi is a city' and q be 'India is a country'. What is the negation of the statement 'Delhi is a city and India is a country'?
By De Morgan's Laws, ~(p ∧ q) ≡ ~p ∨ ~q. The 'and' changes to 'or'.
Which of the following is an example of an INCLUSIVE 'OR'?
Inclusive OR means one, the other, or both. A person can have both a BSc degree and 3 years of experience and still apply.
Ready to test this chapter?
Take the Chapter Test →