Logical Implication - Revision history

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Lila: Created page with "A logical implication is a relationship between two statements. If a statement Q is always true when another statement P is true, then we say that "P implies Q", which is deno..."

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Created page with "A logical implication is a relationship between two statements. If a statement Q is always true when another statement P is true, then we say that "P implies Q", which is deno..."

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A logical implication is a relationship between two statements. If a statement Q is always true when another statement P is true, then we say that "P implies Q", which is denoted symbolically as <math> P \implies Q </math>. Note that if P is false, Q does not necessarily have to be false. For example, if x > 10, then x is also greater than 0, so we can say that "<math> x > 10 \implies x > 0 </math>". However, if x is less than 10, it doesn't necessarily mean that x isn't greater than 0. That is, <math> x > 10 \implies x > 0 </math> does NOT mean that <math> x \le 10 \implies x \le 0 </math>. The truth table for logical implication is as follows:

{| class="wikitable" style="margin:1em auto 1em auto; text-align:center;"
|-
! <math>P</math> || <math>Q</math> || <math>P \Rightarrow Q</math>
|-
| T || T || T
|-
| T || F || F
|-
| F || T || T
|-
| F || F || T
|}

Note that while the inverse of <math> P \implies Q </math> (that is, <math> \neg P \implies \neg Q </math>) is not necessarily true, the contrapositive (<math> \neg Q \implies \neg P </math> is. For example, <math> x > 10 \implies x > 0 </math> and its contrapositive, <math> x \leq 0 \implies x \leq 10 </math>, are both true or both false at the same time for all values of x.

==Resources==
* [https://sites.millersville.edu/bikenaga/math-proof/truth-tables/truth-tables.html Truth Tables, Tautologies, and Logical Equivalences], Millersville University

@@ Line 1: / Line 1: @@
-A logical implication is a relationship between two statements. If a statement <math> Q </math> is always true when another statement <math> P </math> is true, then we say that <math> P </math> implies <math> Q </math>, which is denoted symbolically as <math> P \implies Q </math>. Note that if <math> P </math> is false, <math> Q </math> does not necessarily have to be false. For example, if <math> x > 10 </math>, then <math> x > 0 </math>, so we can say that "<math> x > 10 \implies x > 0 </math>". However, if x is less than 10, it doesn't necessarily mean that x isn't greater than 0. That is, <math> x > 10 \implies x > 0 </math> does NOT mean that <math> x \le 10 \implies x \le 0 </math>. The truth table for logical implication is as follows:
+==Implication==
 {| class="wikitable" style="margin:1em auto 1em auto; text-align:center;"
@@ Line 13: / Line 65: @@
 | F || F || T
 |}
 Note that while the inverse of <math> P \implies Q </math> (that is, <math> \neg P \implies \neg Q </math>) does not necessarily have the same truth value as <math> P \implies Q </math>, the contrapositive (<math> \neg Q \implies \neg P </math>) does. For example, <math> x > 10 \implies x > 0 </math> and its contrapositive, <math> x \leq 0 \implies x \leq 10 </math>, are logically equivalent, and always have the same truth value for any number <math> x </math>.
@@ Line 18: / Line 84: @@
 ==Resources==
 * [https://sites.millersville.edu/bikenaga/math-proof/truth-tables/truth-tables.html Truth Tables, Tautologies, and Logical Equivalences], Millersville University

@@ Line 1: / Line 1: @@
-A logical implication is a relationship between two statements. If a statement Q is always true when another statement P is true, then we say that "P implies Q", which is denoted symbolically as <math> P \implies Q </math>. Note that if P is false, Q does not necessarily have to be false. For example, if x > 10, then x is also greater than 0, so we can say that "<math> x > 10 \implies x > 0 </math>". However, if x is less than 10, it doesn't necessarily mean that x isn't greater than 0. That is, <math> x > 10 \implies x > 0 </math> does NOT mean that <math> x \le 10 \implies x \le 0 </math>. The truth table for logical implication is as follows:
+A logical implication is a relationship between two statements. If a statement <math> Q </math> is always true when another statement <math> P </math> is true, then we say that <math> P </math> implies <math> Q </math>, which is denoted symbolically as <math> P \implies Q </math>. Note that if <math> P </math> is false, <math> Q </math> does not necessarily have to be false. For example, if <math> x > 10 </math>, then <math> x > 0 </math>, so we can say that "<math> x > 10 \implies x > 0 </math>". However, if x is less than 10, it doesn't necessarily mean that x isn't greater than 0. That is, <math> x > 10 \implies x > 0 </math> does NOT mean that <math> x \le 10 \implies x \le 0 </math>. The truth table for logical implication is as follows:
 {| class="wikitable" style="margin:1em auto 1em auto; text-align:center;"
@@ Line 14: / Line 14: @@
 |}
-Note that while the inverse of <math> P \implies Q </math> (that is, <math> \neg P \implies \neg Q </math>) does not necessarily have the same truth value as <math> P \implies Q </math>, the contrapositive (<math> \neg Q \implies \neg P </math>) does. For example, <math> x > 10 \implies x > 0 </math> and its contrapositive, <math> x \leq 0 \implies x \leq 10 </math>, are logically equivalent, and always have the same truth value for any number x.
+Note that while the inverse of <math> P \implies Q </math> (that is, <math> \neg P \implies \neg Q </math>) does not necessarily have the same truth value as <math> P \implies Q </math>, the contrapositive (<math> \neg Q \implies \neg P </math>) does. For example, <math> x > 10 \implies x > 0 </math> and its contrapositive, <math> x \leq 0 \implies x \leq 10 </math>, are logically equivalent, and always have the same truth value for any number <math> x </math>.
 ==Resources==
 * [https://sites.millersville.edu/bikenaga/math-proof/truth-tables/truth-tables.html Truth Tables, Tautologies, and Logical Equivalences], Millersville University

@@ Line 14: / Line 14: @@
 |}
-Note that while the inverse of <math> P \implies Q </math> (that is, <math> \neg P \implies \neg Q </math>) does not necessarily have the same truth value as <math> P \implies Q </math>, the contrapositive (<math> \neg Q \implies \neg P </math>) does. For example, <math> x > 10 \implies x > 0 </math> and its contrapositive, <math> x \leq 0 \implies x \leq 10 </math>, are both true or both false at the same time for all values of x.
+Note that while the inverse of <math> P \implies Q </math> (that is, <math> \neg P \implies \neg Q </math>) does not necessarily have the same truth value as <math> P \implies Q </math>, the contrapositive (<math> \neg Q \implies \neg P </math>) does. For example, <math> x > 10 \implies x > 0 </math> and its contrapositive, <math> x \leq 0 \implies x \leq 10 </math>, are logically equivalent, and always have the same truth value for any number x.
 ==Resources==
 * [https://sites.millersville.edu/bikenaga/math-proof/truth-tables/truth-tables.html Truth Tables, Tautologies, and Logical Equivalences], Millersville University