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Discrete Structure Presentation
IMPLICATION: The statement “p implies q” means that if p is true, then q must also be true. The statement “p implies q” is also written “if p then q” or sometimes “q if p”. Statement p is called the premise of the implication and q is called the conclusion. Symbol: “→” Examples: 1- if you obtain 90% or above marks, then you will get A+ grade. 2- If the U.S. discovers that the Taliban Government is involved in the terrorist attack, then it will retaliate against Afghanistan. 3- x=6 implies x+6= 12
Understanding Example With Truth Table: • “If you get first position, then I’ll give you PS5.” Explanation: • The statement will be true if I keep my promise and false if I don't. • Suppose it's true that you get an First position and it's true that I give you a PS5. Since I kept my promise, the implication is true. This corresponds to the first line in the table. • Suppose it's true that you get first position but it's false that I give you a PS5. Since I didn't keep my promise, the implication is false. This corresponds to the second line in the table. • What if it's false that you get first position? Whether or not I give you a PS5, I haven't broken my promise. Thus, the implication can't be false, so (since this is a two-valued logic) it must be true. This explains the last two lines of the table.
Some Terminology for p → q: • if p, then q (Example given in third slide) • p implies q ( Example given in third slide) • if p, q • p only if q • q if p • q when p • a necessary condition for p is q • q unless ￢p”
Converse,Contrapositive,Inverse: • Converse: Let p → q • The Converse of “p → q” is “p → q”. • Contrapositive: Let p →q • The contrapositive of “p → q” is “¬q→ ¬p”. • Inverse: Let p →q • The Inverse of “p →q” is “¬p → ¬q”
Example: Given an if-then statement "if p , then q ," we can create three related statements: A conditional statement consists of two parts, a hypothesis in the “if” clause and a conclusion in the “then” clause. For instance, “If it rains, then they cancel school.” "It rains" is the hypothesis."They cancel match"is the conclusion. • To form the converse of the conditional statement, interchange the hypothesis and the conclusion. The converse of "If it rains, then they cancel match"is "If they cancel school, then it rains." • To form the inverse of the conditional statement, take the negation of both the hypothesis and the conclusion. The inverse of “If it rains, then they cancel match”is “If it does not rain, then they do not cancel match.” • To form the contrapositive of the conditional statement, interchange the hypothesis and the conclusion of the inverse statement. The contrapositive of "If it rains, then they cancel match"is "If they do not cancel match, then it does not rain."
Biconditional Statement: Definition: The Biconditional statement is defined to be true when both parts(p & q) have same truth value. The biconditional p,q represent “p if and only if q”, where p is hypothesis and q is conclusion. Symbol: “↔” Explanation: p ↔ qmeans that P and Q are equivalent. So the implication is true if P and Q are both true or if P and Q are both false; otherwise, the double implication is false.
Other ways to express P ↔Q: • p is necessary and sufficient for q • if p then q, and conversely • p iff q.
Examples: Example#1: P: x+ 8= 10 , Q: x=2 P ↔Q =? Sol: The biconditonal p,q represents the sentence: "x + 8 = 10 if and only if x = 2." When x = 2, both a and b are true. When x is not equal to2, both a and b are false. A biconditional statement is defined to be true whenever both parts have the same truth value. Example#2: A= a polygon is a triangle , B= A polygon has exactly 3 sides , A ↔B=? Sol: “A polygon is a triangle if and if only it has exactly three sides”. • Note that in the biconditional above, the hypothesis is: "A polygon is a triangle" and the conclusion is: "It has exactly 3 sides." It is helpful to think of the biconditional as a conditional statement that is true in both directions
