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Mathematical Reasoning - Special Words/Phrases (And/Or, Quantifiers)

Grade 11CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Compound Statements and Connectives: A compound statement is a mathematical statement formed by combining two or more simple statements using words like 'And', 'Or', 'If-then', and 'Only if'. Visually, imagine two individual logical blocks pp and qq linked by a bridge representing the connective.

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The Connective 'And': A compound statement p∧qp \wedge q is true only if both component statements pp and qq are true. If even one is false, the entire compound is false. Visually, this represents the intersection of two sets P∩QP \cap Q in a Venn diagram, where only the overlapping region is considered.

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The Connective 'Or' (Inclusive): The 'Or' is inclusive if the compound statement p∨qp \vee q is true when pp is true, qq is true, or both are true. In a Venn diagram, this is visualized as the union of two sets P∪QP \cup Q, covering all regions within both circles.

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The Connective 'Or' (Exclusive): The 'Or' is exclusive if the compound statement is true when exactly one of the components is true, but not both. For example, 'A person is at home or at the office'. Visually, this is the area of the two circles in a Venn diagram excluding the middle intersection part.

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Universal Quantifiers: These are phrases like 'For all' or 'For every', denoted by the symbol ∀\forall. A statement with a universal quantifier is true only if the condition holds for every single element in the given domain. Visually, imagine a group of objects where a specific property is applied to the entire collection without exception.

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Existential Quantifiers: These are phrases like 'There exists' or 'For some', denoted by the symbol ∃\exists. A statement with an existential quantifier is true if there is at least one element in the domain that satisfies the condition. Visually, this is represented by identifying at least one specific instance or point within a set that meets the requirement.

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Negation of Quantified Statements: To negate a statement containing 'For all', we change it to 'There exists... such that not'. To negate 'There exists', we change it to 'For all... not'. Visually, the negation of a fully shaded set (All) is the existence of at least one empty point (Some not).

📐Formulae

p∧qp \wedge q (True only if both pp and qq are True)

p∨qp \vee q (True if at least one of pp or qq is True)

∼(p∧q)≡(∼p)∨(∼q)\sim (p \wedge q) \equiv (\sim p) \vee (\sim q)

∼(p∨q)≡(∼p)∧(∼q)\sim (p \vee q) \equiv (\sim p) \wedge (\sim q)

∼[∀x∈S,p(x)]≡∃x∈S,∼p(x)\sim [\forall x \in S, p(x)] \equiv \exists x \in S, \sim p(x)

∼[∃x∈S,p(x)]≡∀x∈S,∼p(x)\sim [\exists x \in S, p(x)] \equiv \forall x \in S, \sim p(x)

💡Examples

Problem 1:

Identify the component statements and check the truth value of the compound statement: '1010 is a multiple of 22 and 55'.

Solution:

  1. Identify component statements: pp: 1010 is a multiple of 22 (True because 2×5=102 \times 5 = 10) qq: 1010 is a multiple of 55 (True because 5×2=105 \times 2 = 10)
  2. The connective used is 'And'.
  3. Since both pp and qq are true, the compound statement p∧qp \wedge q is True.

Explanation:

A compound statement with 'And' requires every component to be true for the whole statement to be true.

Problem 2:

Write the negation of the statement: 'There exists a real number xx such that x2+1=0x^2 + 1 = 0'.

Solution:

  1. Identify the quantifier: The statement uses the existential quantifier 'There exists' (∃\exists).
  2. Apply the rule for negation: Change 'There exists' to 'For all' (∀\forall) and negate the condition.
  3. The condition is x2+1=0x^2 + 1 = 0. Its negation is x2+1≠0x^2 + 1 \neq 0.
  4. Negated Statement: 'For every real number xx, x2+1≠0x^2 + 1 \neq 0'.

Explanation:

Negating an existential quantifier involves switching to a universal quantifier and negating the inner property.