Mathematical Reasoning and Statistics
Mathematical Reasoning and Statistics tests truth values of statements, connectives, quantifiers, validity of arguments, and measures of dispersion. The statistics part usually asks for mean deviation, variance, standard deviation, and coefficient of variation.
Core rules
A statement is a sentence that is either true or false, not both. Negation reverses truth value. Conjunction p∧q is true only when both are true; disjunction p∨q is false only when both are false. Implication p→q is false only when p is true and q is false. Its contrapositive is ∼q→∼p and is logically equivalent; the converse q→p is not. Quantifiers: 'for all' and 'there exists'. For data, variance σ²=Σfᵢ(xᵢ−x̄)²/N, standard deviation σ=√σ², and coefficient of variation CV=(σ/x̄)×100%.
Worked example
For 2,4,6,8,10, mean x̄=6. Deviations are −4,−2,0,2,4; squares sum to 16+4+0+4+16=40. Variance=40/5=8; SD=√8=2√2. CV=(2√2/6)×100%≈47.14%.
Traps
The converse of p→q is not equivalent to it; only the contrapositive is. 'Or' in mathematics is usually inclusive unless 'exclusive or' is stated. In variance, use the correct N for the given data and do not mix ungrouped and grouped formulas. CV is a percentage and is unitless, so do not attach the original units. A high SD does not by itself mean the mean is unreliable; compare CV.