CONJUNCTION

Definition : A conjunction is a compound statement formed by joining two statements with the connector AND. The conjunction "p and q" is symbolized by p ˄ q. A conjunction is true when both of its combined parts are true; otherwise it is false. With a conjunction, both statements must be true for the conjunction to be true.

Example 1: Statement p represents the sentence, "Ann is on the softball team," and  statement q represents the sentence, "Paul is on the football team." The symbol ˄  is a logical connector which means "and." Therefore, the compound statement  p ˄ q represents     the sentence, "Ann is on the softball team and Paul is on  the footballteam." The statement p ˄ q is a conjunction. p q p ˄ q T T T T F F F T F F F F Example 2: Given: a: A square is a quadrilateral. b: Harrison Ford is an American actor. Problem:  Construct a truth table for the conjunction "a and b." Solution: a b a ˄ b T T T T F F F T F F F F Example 3: Given: r: The number x is odd. s: The number x is prime. Problem: Can we list all truth values for r ˄ s in a truth table?Why or why not? Solution:  If x = 3, then r is true, s is true. The conjunction r ˄ s is true. If x = 9, then r is true, s is false. The conjunction r ˄ s is false. If x = 2, then r is false, s is true. The conjunction r ˄ s is false. If x = 6, then r is false, s is false. The conjunction r ˄ s is false.

IMPLICATION

Definition : An implication, symbolized by p→q, is an if-then statement in which p  is a hypothesis and q is a conclusion. The logical connector in a conditional statement is denoted by the symbol →. The conditional is defined to be true unless  a true hypothesis leads to a false conclusion. A truth table for p→q is shown below. Example 1: Given: p: I do my homework. q: I get my allowance. Problem: What does p→q represent? Solution: In Example 1, p represents, "I do my homework," and q represents "I get my  allowance." The statement p→q is a conditional statement which represents  "If p, then q. p q p→q T T T T F F F T T F F T Example 2: Given: a: The sun is made of gas. b: 3 is a prime number. Problem: Write a→b as a sentence. Then construct a truth table for this conditional. Solution: The conditional a→b represents "If the sun is made of gas, then  3 is a prime number." a b a→b T T T T F F F T T F F T Example 3: Given: r: 8 is an odd number. false s: 9 is composite. true Problem: What is the truth value of  r→s? Solution:Since hypothesis r is false and conclusion s is true,the implication r→ s

BICONDITIONAL

Definition:

A biconditional statement is defined to be true whenever both parts have the same truth value. The biconditional operator is denoted by a double-headed arrow ↔. The biconditional p↔q represents "p if and only if q," where p is a hypothesis and q is a conclusion. The following is a truth table for biconditional p↔q. In the truth table above, p↔q is true when p and q have the same truth values, (i.e., when either both are true or both are false.) p q p↔q T T T T F F F T F F F T

Example 1:
Given: p: A polygon is a triangle. q: A polygon has exactly 3 sides. Problem: What does the statement p↔q represent?
Solution:The statement p↔q represents the sentence, "A polygon is a triangle if and only if it has exactly 3 sides."

Example 2:

Given:	a: x + 2 = 7
	b: x = 5
Problem:	Write a↔b as a sentence. Then determine its truth values a↔b.

Solution: The biconditonal a↔b represents the sentence: "x + 2 = 7 if and only if x = 5." When x = 5, both a and b are true. When x ↔5, both a and b are false. A biconditional statement is defined to be true whenever both parts have the same truth value. Accordingly, the truth values of a↔b are listed in the table below.

a	b	a↔b
T	T	T
T	F	F
F	T	F
F	F	T

Example 3:

Given:	x: I am breathing
	y: I am alive
Problem:	Write x↔y as a sentence.

Solution:  x↔y represents the sentence, "I am breathing if and only if I am alive."