Basic Mathematical Logic II for IITJEE and Maths and Physics Olympiads

Reference: Introduction to General Topology by K. D. Joshi

Logical Implications

As far as mathematics is concerned, the most important aspect of logic is the statements of the form ‘if …then…’. These are often called implications. Theorems in mathematics and physics are commonly expressed as statements of this form or they can be paraphrased in such a form. For example, the theorem that if two triangles are congruent, then they are also similar. Another example: the sum of the three interior angles of a triangle is 180 degrees. This statement can be paraphrased as ‘if ABC is a triangle, then the sum of its three interior angles is 180 degrees’. Statements of the form ‘if p…then q’ are called logical implications or just implications. The statement p is called the hypothesis and q is called the conclusion.

The following are the various forms of an implication statement:

  1. p implies q
  2. q follows from p.
  3. q is a (logical) consequence of p.
  4. If p is true, then q is true.
  5. If q is false, then p is false.
  6. p is false, unless q holds.
  7. p is a sufficient condition for q.
  8. q is a necessary condition for p.
  9. p is true, only if q is true.

In mathematics, it often happens that we combine together an implication statement along with its converse. For example, take the well-known theorem, ‘the sum of the opposite angles in a cyclic quadrilateral is 180 degrees and conversely’. Such a statement is called a double implication. That is, if p, then q AND if q, then p. It can be written as p \Longleftrightarrow q. The following are the variations of a double implication:

  1. p and q imply each other.
  2. p and q are equivalent to each other.
  3. p holds if and only if q holds.
  4. q is a characterization of p.
  5. q holds if p does and conversely.
  6. q holds if p does, but not otherwise.
  7. If p is true, then q is true and if p is false, so is q.
  8. q is a necessary as well as a sufficient condition for p.

In the double implication p \Longleftrightarrow q, the implication p \Longrightarrow q is called the direct implication or the ‘only if’ part and the other implication q \Longrightarrow p is called the converse implication or the if part of the theorem.

More later,

Nalin Pithwa

 

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