Logical Reasoning

These properties describe how mathematical operations behave. Closure means when you combine two numbers using an operation, you get another number in the same set. For instance, adding two whole numbers always gives a whole number. Commutativity means order doesn't matter: 3 + 5 equals 5 + 3. Associativity lets you group numbers differently without changing results: (2 + 3) + 4 equals 2 + (3 + 4). Distributivity connects multiplication and addition: 3(2 + 4) equals 3(2) + 3(4).

Identity elements are special numbers that don't change others. Zero is the additive identity because any number plus zero stays the same. One is the multiplicative identity because any number times one remains unchanged. Inverses are opposites that combine to give the identity. The additive inverse of 5 is –5 because 5 + (–5) = 0.

Logical reasoning means checking whether statements make sense and whether conclusions follow properly from what we know. Think of it like investigating a case: if all students in your school wear uniforms, and Chioma is a student in your school, then Chioma must wear a uniform. That's deduction—drawing a certain conclusion from given facts.

A statement is either true or false, nothing in between. The rule of syntax means following the proper structure of logical arguments. Implications show "if-then" relationships: if it rains, then the ground gets wet. Arguments are valid only when the conclusion necessarily follows from the premises given.

Consider this Nigerian example: All successful traders understand mathematics; Adekunle is a successful trader; therefore, Adekunle understands mathematics. This is valid reasoning because the conclusion logically follows.

A truth table is simply a chart that shows all possible outcomes when you combine logical statements. Think of it like a market vendor listing every possible combination of goods and their prices. For each combination of inputs marked as true (T) or false (F), you determine what the final answer should be.

The most common operations are AND, OR, and NOT. With AND, both statements must be true for the result to be true—like saying "I'll go to the party AND bring my friend." With OR, only one needs to be true. NOT simply flips the value: if something is true, NOT makes it false, and vice versa.

Consider this Nigerian example: "It rains AND the road floods." The statement is only true when both conditions happen together. If it rains but the road doesn't flood, that's false. Your truth table would show all four possibilities.

Think of a function like a money transfer service. The domain is the set of all possible inputs—the money you're allowed to send. The co-domain is the complete set of possible outputs where your money could land.

For example, imagine a function that converts Nigerian naira to dollars. Your domain might be amounts from ₦1,000 to ₦1,000,000. The co-domain would be all possible dollar values the system accepts, say $0 to $3,000. However, the range (which is different) is only the actual dollar amounts you receive based on the conversion rate.

The key difference: co-domain is what *could* happen, range is what *actually* happens. A student often confuses these two concepts during exams.

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A mapping is like a function that connects one set to another. Think of it like matching students in your class to their phone numbers—each student gets exactly one number.

A one-to-one mapping means different inputs give different outputs. If students A and B are different, their phone numbers must be different too. An onto mapping means every phone number in the target set belongs to at least one student—no number is left unmapped. When you combine both properties, that's a bijection.

An identity mapping sends each element to itself. If your set is {1, 2, 3}, the identity mapping keeps them as {1, 2, 3}. A constant mapping, however, sends every element to the same single value. For example, mapping all students to the number 100 regardless of who they are.

Consider a Nigerian example: assigning registration numbers to JAMB candidates. Each candidate gets one unique number (one-to-one), every number in the system is assigned (onto), so it's bijective.

When you have a function that takes input and produces output, the inverse function does the exact opposite—it takes the output and returns you to the original input. Think of it like a journey: if a function moves you from Lagos to Abuja, its inverse brings you back from Abuja to Lagos.

Mathematically, if function f maps x to y, then the inverse function f⁻¹ maps y back to x. For example, if a Nigerian trader's profit function is P(x) = 5x + 1000, where x is items sold, the inverse would help you find how many items were sold given a particular profit amount.

To find an inverse, swap x and y in your equation, then solve for the new y. Remember that not all functions have inverses—the original function must be one-to-one (each input gives one unique output).

When you have two functions, you can combine them to create a new function called a composite function. Think of it like two operations happening one after another. If you have function f and function g, the composite function (f∘g)(x) means you apply g first, then apply f to the result you get.

Picture a Nigerian trader calculating profit. Let g(x) represent the cost of goods, and f(x) represents the profit margin added to that cost. When you find (f∘g)(x), you're finding the final selling price by first calculating the cost, then adding profit to it. The order matters greatly—doing it backwards gives a different answer.

Remember that (f∘g)(x) is not the same as (g∘f)(x). Always work from the inside function outward, substituting carefully and simplifying your answer completely.

A linear function is simply a mathematical relationship between two variables that produces a straight line when graphed. Think of it like calculating transport fares in Lagos: if a bike rider charges ₦100 base fare plus ₦50 per kilometre, your total cost becomes a linear function of distance travelled. The equation would be Cost = 100 + 50x, where x represents kilometres.

Linear equations are statements showing that two expressions are equal, like 2x + 3 = 11. You solve them by isolating the variable. Linear inequalities work similarly, but instead of equals signs, they use symbols like < (less than), > (greater than), ≤ or ≥. When solving inequalities, remember that multiplying or dividing by negative numbers flips the inequality sign—this trips up many students.