Concept 01 / 04Mathematical Reasoning
Statements, Negation and Compound Statements with And and Or
What counts as a mathematical statement, truth values, negation, conjunction (and) and disjunction (inclusive or), truth tables, exclusive or, and negating compound statements correctly.
A statement (proposition) is a declarative sentence that is either true or false, but not both. ' is a prime number' is a true statement; '' is a false statement. The following are not statements: questions ('Is it raining?'), commands ('Close the door'), exclamations, and sentences whose truth depends on an unspecified variable or on opinion ('', 'Mathematics is interesting'). Open sentences with variables become statements once the variable is fixed or quantified (see the concept on quantifiers).
Statements are denoted by letters , and their truth values by T (true) and F (false).
Negation. The negation ('not ') is true exactly when is false. Negating a statement requires care with its wording: the negation of 'All students passed' is 'Some student did not pass' (or 'Not all students passed'), not 'No student passed'. The negation of ' is rational' is ' is not rational'.
Conjunction. (' and ') is true only when both and are true. Words like 'but', 'although' and 'yet' also express conjunction logically. Caution: not every 'and' in English joins two statements. 'A mixture of alcohol and water can be separated' is a single statement about a mixture.
Disjunction. (' or ') is true when at least one of is true; it is false only when both are false. This is the inclusive or. The exclusive or ('either or but not both'), written or , is true when exactly one is true. In mathematics, 'or' means inclusive unless stated otherwise; in everyday language ('tea or coffee?') it is often exclusive, and the context decides.
Truth tables. With component statements there are combinations of truth values, so a truth table has rows. The figure shows the table for , and .
Negating compound statements (De Morgan's laws).
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In words: 'not both' means 'at least one fails'; 'not either' means 'both fail'. For example, the negation of ' is even and is odd' is ' is not even or is not odd'. These are the logic versions of De Morgan's laws for sets, with , and complement.
Key Takeaways:
- A statement is definitely true or false; questions, commands and opinions are not statements.
- reverses the truth value.
- : true only if both are true. : false only if both are false.
- Mathematical 'or' is inclusive.
- components: rows.
- ; .
- Pitfall: negating 'all' as 'none'; the correct negation is 'not all' (some are not).
Brain-Teaser 1: Which of the following are statements? (a) 'The sum of the angles of a triangle is .' (b) '.' (c) 'Every even number greater than is the sum of two primes.' (d) 'Please solve this problem.'
Solution: (a) is a true statement. (b) is an open sentence: its truth depends on , so it is not a statement until is specified or quantified. (c) is a statement (Goldbach's conjecture): it is definitely either true or false, even though nobody currently knows which. (d) is a request, not a statement. The criterion is 'has a definite truth value', not 'we know the truth value'.
Brain-Teaser 2: Write the negation of 'Every natural number is greater than and some integer is negative'.
Solution: By De Morgan, the negation is 'Some natural number is not greater than or every integer is non-negative'. Each part is negated with its quantifier switched ('every' becomes 'some', and 'some' becomes 'every'), and the 'and' becomes 'or'. The original statement is true, so its negation is false, as a quick check.
Brain-Teaser 3: If is true and is false, find the truth values of , and .
Solution: is T, so is T. is F and is F, so is F. is T, so its negation is F. Evaluating from the innermost parts outwards, exactly as with arithmetic expressions, avoids mistakes.
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