MAT-230: Discrete Mathematics

MAT-230 at Southern New Hampshire University studies mathematical structures that are discrete rather than continuous.

The catalog lists operations on sets, logic, truth tables, counting, relations and digraphs, functions, trees and graph theory, and says a significant goal is to improve critical thinking and problem solving.

Students find it different from algebra and calculus because the emphasis is on reasoning and precise definitions. A correct answer is often a short argument, so learning to write clear steps matters as much as getting the final result.

What MAT-230 Covers

Key Concepts Explained

Truth Tables and Implication

The statement "if p then q" is false only when p is true and q is false. That surprises many students, because a conditional with a false hypothesis counts as true.

Counting with Combinations

The number of handshakes among n people is the number of ways to choose 2, written C(n,2) = n(n - 1)/2.

Graph Facts

In any graph, the sum of the vertex degrees equals twice the number of edges, because each edge adds one to the degree of two vertices. A tree on n vertices always has n - 1 edges.

Typical Assignments and How to Approach Them

Writing a Short Argument

Many discrete maths marks are for reasoning, not computation. A good short proof states the claim, restates the relevant definition, and then moves in small, justified steps to the conclusion. Avoid jumping to results you have not shown.

When asked to disprove a statement, one counterexample is enough. Choose a small, clear one and check it against every part of the statement.

A Worked Relations Example

Let R on the set {1, 2, 3} be R = {(1,1), (2,2), (3,3), (1,2), (2,1)}. Check each property against the definition.

• Reflexive: (1,1), (2,2) and (3,3) are all present, so yes.
• Symmetric: (1,2) and (2,1) both appear, so yes.
• Transitive: (1,2) and (2,1) force (1,1), and (2,1) and (1,2) force (2,2); both are present, so yes.

A Sets Reminder

For A = {1, 2, 3} and B = {2, 3, 4}, the union is {1, 2, 3, 4}, the intersection is {2, 3} and A minus B is {1}. A set with n elements has 2^n subsets, so A has 8.

Arguments and Logical Equivalence

Two statements are logically equivalent when their truth tables match in every row. A classic pair is the conditional and its contrapositive: if p then q is equivalent to if not q then not p. De Morgan's laws turn not (p and q) into (not p) or (not q).

Use these equivalences to simplify statements and to prove claims by contrapositive. Writing both truth tables side by side is a reliable way to check an equivalence you suspect.

Where Students Get Stuck

• Implication. Remember that false hypotheses make a conditional true.
• Counting order. Mixing permutations and combinations is the most common slip.
• Relation properties. Test with a concrete pair, not only a general feeling.
• Vague proofs. Write each step as a full sentence that cites a definition.

Study Tips for MAT-230

• Re-learn each definition word for word, then restate it in your own words.
• Draw a diagram for every relation, function or graph question.
• Test claims on small examples before trying to prove them.
• Keep a short list of counting rules with a trigger phrase for each.

How We Help with MAT-230

Send the problems, the lesson notes and your working. A maths tutor can explain the reasoning, prepare worked solutions, or review your proofs for gaps. For the computing side of these ideas, see the related data structures page.

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