Implies

From CS2800 wiki

If [math]P [/math] and [math]Q [/math] are propositions, then "if [math]P [/math] then [math]Q [/math]" (written [math]P \href{/cs2800/wiki/index.php?title=%5Cimplies&action=edit&redlink=1}{\implies} Q [/math] or "[math]P [/math] implies [math]Q [/math]") is a proposition. It is true if either [math]P [/math] is false, or if [math]Q [/math] is true.

  • To prove "if [math]P [/math] then [math]Q [/math]", assume [math]P [/math] and then prove [math]Q [/math].
  • If you know "if [math]P [/math] then [math]Q [/math]", and you also know [math]P [/math], you can conclude [math]Q [/math]. This technique is sometimes referred to as "modus ponens".
  • To disprove "if [math]P [/math] then [math]Q [/math]", you must show that [math]P [/math] is true and that [math]Q [/math] is false ("[math]P [/math] implies [math]Q [/math]" only makes a claim about the world where [math]P [/math] is true; an example where [math]P [/math] is false doesn't contradict the claim).