MATHEMATICAL STATEMENTS 1 Statement - defined as a declarative sentence - either true or false > Truth Value - Truthfulness or falseness of a mathematical statement EXAMPLE: p: Mr. Dalde is a grade 9 teacher TRUE q: Leni R. is the president of the Philippines FALSE > Open Sentences - One that contain on or more variables or pronouns EXAMPLE: She is a Theresian X + 7 = 15 ? Negation of Statement - Directly denies the claim of the original statement - ~ p read as "not p" - Not, does not, do not - Affects the truth value - = & #, > & >, < & >, < & >, > & < EXAMPLE: p: Mr. Dalde is a grade 9 teacher TRUE ~p: Mr Dalde is not a grade 9 teacher FALSE p: Leni R. is the president of the Philippines FALSE ~p: Leni R. is not the president of the Philippines TRUE p: 200 + 53 # 253 FALSE ~p: 200 + 53 = 253 TRUE COMPOUND STATEMENTS > Compound Statement - Formed by joining two or more simple statements - And, or, if .... Then, if and only if a) Conjunction - and & denoted by A b) Disjunction - or & denoted by V GIVEN: p: Martin is a pizza lover q: Cheska is a quesadilla lover ~p: Martin is not a pizza lover. ~q: Cheska is not a quesadilla lover. p A q: Martin is a pizza lover and Cheska is a quesadilla lover p V q: Martin is a pizza lover or Cheska is a quesadilla lover ~q A p: Cheska is not a quesadilla lover and Martin is a pizza lover. ~p v q: Martin is not a pizza lover or Cheska is a quesadilla lover. c) Conditional (Implantation) - if .... then & p-›q d.) Converse - if .... Then & q->p e.) Inverse - negation of conditional ~p->~q f.) Contrapositive - negation of Converse ~q->~p GIVEN: p: You are studying in STCQC. q: You are a woman p->q: If you are studying in STCQC, then you are a woman TRUE
q->p: If you are a woman, then you are studying in STCQC FALSE ~p->~q: If you are not studying in STCQC, then you are not a woman FALSE ~q->~p: If you are not a woman, then you are not studying in STCQC TRUE · Equivalent Statements - statements that are written differently but hold the same logical equivalence - Conditional = Contrapositive & Converse = Inverse f) Biconditional - conjunction of a conditional statement and its converse - (p->q) ^ (q->p) - if and only if p <> q, GIVEN: p: A number is even. q: A number is divisible by two. p <> q: A number is even if and only if it is divisible by two. TRUE (p->q) ^ (q->p): If a number is even then the number is divisible by two and if the number is divisible by two then the number is even TRUE