necessary and sufficient conditions in logic statements

Searching for precise definitions is employing the necessary and/or sufficient conditions for the applications of a term, concept, or occurrence of a phenomenon or event.

It is important to determine if the conditions in the hypothesis are necessary/sufficient to justify conclusion.

Sufficient and necessary conditions

These help to identify the terms hypothesis and conclusion in a conditional statement. The terms sufficient and necessary in English mean exactly what they mean in conditional logic.

If p and q denote statements then: "p is a sufficient condition for q". Meaning that the occurrence of p suffices to guarantee the occurrence of q.

"p is a necessary condition for q". meaning that the occurrence of p is necessary to obtain the occurrence of q.

These are not equivalent to each other. They exist as converses of each other, meaning if p, then q is equivalent to:

• p is a sufficient condition for q
• q is a necessary condition for p

Note: It is possible for one to be sufficient and another to be not necessary, or both to be sufficient/necessary to each other, etc.

"p if and only if, q" is equivalent:

• p is a necessary and sufficient condition for q
• q is a sufficient and necessary condition for p

Interpreting a sufficient condition

Statement to rewrite: "For a function to be continuous, it is sufficient that it is differentiable"

Note: There are multiple ways to write this statement. Sufficient is not fixed in its position.

• The truth of the condition "a function being differentiable" is sufficient to ensure

the truth of the condition "the function being continuous"

Equivalent form: "If a function is differentiable, then a function is continuous"

Interpreting a necessary condition

Statement to rewrite: "Doing homework regularly is a necessary condition for Jim to pass the course"

• The truth of the condition "Jim doe shomework regularly" is necessary for the

condition "Jim to pass the course" to be true

Equivalent form: "If Jim passes the course, then he does the homework regularly"

Contrapositive form: "If Jim doe snot do homework regularly, then he does not pass the course"

Interpreting a sufficient and necessary condition

Statement to rewrite: "Being a number divisible by 6 is a sufficient and necessary condition for it being divisible by both 2 and 3"

• The truth of the condition "a number being divisible by 6" is sufficient to ensure

the truth of the condition "a number being divisible by both 2 and 3", and the truth of the condition "a number being divisible by 6 is necessary for the truth of the condition "a number being divisible by both 2 and 3"

Equivalent form: "A number is divisible by 6 if, and only if, it is divisible by both 2 and 3"