Chapter 1 Introduction In this chapter we hope to introduce students to applying logical reasoning to prove the validity of economic conclusions(B)from well-defined premises (A). For example, A may be the statement " An allocation-price pair (, p) is a Walrasian equilibrium"and B the statement "the allocation is Pareto efficient". In general, statements such as A and or B may be true or false 1.1 Rules of logic In many cases, we will be interested in establishing the truth of statements of the form"If A, then B. Equivalently, such a statement can be written as:“A→B”;“ A implies”;“ A only if E”;“ a is sufficient for B”;or“Bis for A "Applied to the example given in the previous paragraph, "If A, then B"is just a statement of the First Fundamental Theorem of Welfare Economics. In other cases. we will be interested in the truth of statements of the form"A if and only if B. equivalently, such a statement can be written A”;“ A is necessary and sufficient for B";or“ A Is equivalent to2’ plies “A→ B and B→A” which is just“AB”;“ A implies盛 and B implies Notice that a statement of the form“A→B” is simply a construct of two simple statements connected by =>" Proving the truth of the statement A=B captures the essence of mathematical reasoning; we take the truth of A as given and then establish logically the truth of B follows. Before actually setting out on that path, let us define a few terms. A Theorem on Proposition is a statement that we prove to be true. A Lemma is a theorem we use to prove another theorem. A Corollary is a theorem whose proof isChapter 1 Introduction In this chapter we hope to introduce students to applying logical reasoning to prove the validity of economic conclusions (B) from well-defined premises (A). For example, A may be the statement ìAn allocation-price pair (x, p) is a Walrasian equilibriumî and B the statement ì the allocation x is Pareto efficientî. In general, statements such as A and/or B may be true or false. 1.1 Rules of logic In many cases, we will be interested in establishing the truth of statements of the form ìIf A,then B.î Equivalently, such a statement can be written as: ìA ⇒ Bî; ìA implies Bî; ìA only if Bî; ìA is sufficient for Bî; or ìB is necessary for A.î Applied to the example given in the previous paragraph, ìIf A,then Bî is just a statement of the First Fundamental Theorem of Welfare Economics. In other cases, we will be interested in the truth of statements of the form ìA if and only if B.î Equivalently, such a statement can be written: ìA ⇒ B and B ⇒ Aî which is just ìA ⇔ Bî; ìA implies B and B implies Aî; ìA is necessary and sufficient for Bî; or ìA is equivalent to B.î Notice that a statement of the form ìA ⇒ Bî is simply a construct of two simple statements connected by ì⇒î. Proving the truth of the statement ìA ⇒ Bî captures the essence of mathematical reasoning; we take the truth of A as given and then establish logically the truth of B follows. Before actually setting out on that path, let us define a few terms. A Theorem or Proposition is a statement that we prove to be true. A Lemma is a theorem we use to prove another theorem. A Corollary is a theorem whose proof is 13