Chapter 2 Vocab

Conditional

A conditional statement, symbolized by pq, 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 pq is shown below.

Counterexample

A counterexample is a special kind of example that disproves a statement or proposition. Counterexamples are often used in math to prove the boundaries of possible theorems. In algebra, geometry, and other branches of mathematics, a theorem is a rule expressed by symbols or a formula.

Deductive Reasoning

Deductive reasoning is a logical process in which a conclusion is based on the concordance of multiple premises that are generally assumed to be true. Deductive reasoning is sometimes referred to as top-down logic. Its counterpart, inductive reasoning, is sometimes referred to as bottom-up logic.

Biconditional

In logic, a biconditional is a compound statement formed by combining two conditionals under "and." Biconditionals are true when both statements (facts) have the exact same truth value. A biconditional is read as "[some fact] if and only if [another fact]" and is true when.

Conjecture

In mathematics, a conjecture is a mathematical statement which appears to be true, but has not been formally proven. A conjecture can be thought of as the mathematicians way of saying "I believe that this is true, but I have no proof yet". A conjecture is a good guess or an idea about a pattern.

Proof

In mathematics, a proof is a deductive argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms, along with accepted rules of inference.

Theorem

In mathematics, a theorem is a statement that has been proved on the basis of previously established statements, such as other theorems—and generally accepted statements, such as axioms.

Hypothesis

Math Goodies Glossary. > Glossary > "H" Index. hypothesis. A hypothesis is the "if" part (antecedent) of a conditional statement. View our Lesson on Conditional Statements.

Converse

Switching the hypothesis and conclusion of a conditional statement. For example, the converse of "If it is raining then the grass is wet" is "If the grass is wet then it is raining."

Contrapositive

The contrapositive of a conditional statement is formed by negating both the hypothesis and the conclusion, and then interchanging the resulting negations. In other words, the contrapositive negates and switches the parts of the sentence. It does BOTH the jobs of the INVERSE and the CONVERSE.

Law of Detachment

The law of detachment has a prescribed pattern. There are two premises (statements that are accepted as true) and a conclusion. They must follow the pattern as shown below. This is what is called a valid logical argument.

Law of Syllogism

The law of syllogism takes two conditional statements and forms a conclusion by combining the hypothesis of one statement with the conclusion of another. Here is the general form: P → Q. Q → R.

Conclusion

conditional statement after then. For example, the conclusion of "If a line is horizontal then the line has slope 0" is "the line has slope 0". See also. Hypothesis, converse, inverse, contrapositive, inverse of a conditional, slope.

Inverse

contexts in mathematics the term inverse indicates the opposite of something. This word and its derivatives are used widely in mathematics, as illustrated below. Inverse element of an element x with respect to a binary operation * with identity element e is an element y such that x * y = y * x = e.